REVIEW
A Holographic Dual of Hydrodynamics
High Energy Physics - Theory · Submitted 19 Jul 2006 · 19 references
Pith's verdict: The viscous extension of Janik-Peschanski is real and useful; the abstract's claim that gravity 'determines' the hydrodynamic integration constant is a framing stretch, not a result.
T0 · AI review
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Referee report
The paper
The paper claims
The paper constructs the five-dimensional gravity dual of a strongly interacting plasma that expands in one direction while carrying shear viscosity, and reads off from the horizon of that geometry the one constant the fluid equations alone
The reading
What carries the argument
The machinery is the Fefferman-Graham expansion: write the five-dimensional metric as $ds^2 = (r_0^2 g_{mu nu}(tau,z) dx^mu dx^nu + dz^2)/z^2$ with $g_{mu nu} = g^{(0)}_{mu nu} + z^2 g^{(2)}_{mu nu} + z^4 g^{(4)}_{mu nu} + ...$, identify $g^{(4)}$ with the boundary energy-momentum tensor, and solve the Einstein equations with negative cosmological constant recursively for the higher coefficients. The late-time limit is taken holding $v = z tau^{1/3}$ fixed rather than $z$ fixed, because the horizon moves outward as $z_0 sim tau^{1/3}$ and a fixed-$z$ limit would describe a region infinitely far from it. Resumming the resulting power series in $v$ produces the closed-form metric, and the horizon of that metric is what converts a boundary hydrodynamic statement into a thermodynamic one.
Extended reading notes
Core claim
The central claim is that the late-time bulk geometry dual to a viscous, boost-invariant flow takes the compact form of Eq. (3.17): an asymptotically AdS metric whose components are built from the energy density $rho = rho_0/tau^{4/3} - 2eta_0/tau^2$ and a small parameter $gamma = eta_0/(rho_0 tau^{2/3})$ that measures the viscous correction. The authors show this metric solves the five-dimensional Einstein equations to the stated order in $gamma$, and that the horizon sits at $z_0 = (3/rho)^{1/4}$, so the horizon location is fixed by the energy density alone even though the energy-momentum tensor itself cannot be written that way. From the horizon they obtain the Hawking temperature and hence $rho = (3/8)pi^2 N_c^2 T^4$, and the entropy per unit rapidity and transverse area $S = (N_c^2/2pi)^{1/4}(pi/3)^{3/4} 2sqrt{2} rho_0^{3/4}(1 - (3/2)eta_0/(rho_0 tau^{2/3}) + ...)$. The $tau to infinity$ limit of that expression is an integration constant of the hydrodynamic equations, and the geometry determines it in terms of $rho_0$; the paper checks that it matches the thermodynamic relation $S_infty = (4/3) rho_0 T_0$ obtained by equating the entropy-creation rates computed on the two sides. A second thread asks whether requiring the curvature invariant $(R_{MNKL})^2$ to stay finite at the horizon constrains how the viscosity may fall with time; the answer is that the geometry stays regular for $1/3 <= beta < 5/3$, a window that comfortably contains the physical case $beta = 1$.
Load-bearing premise
The paper assumes that the static relation $eta propto T^3$, known for strongly coupled $N=4$ SYM at finite temperature, continues to hold in a rapidly expanding, time-dependent flow, which is what licenses setting $eta = eta_0/tau$ and thereby produces every viscous term in the metric.
Editorial extensions
If this is right
- The entropy per unit rapidity at asymptotically late time is no longer a free integration constant: it is fixed by the initial energy density through $S_infty = (N_c^2/2pi)^{1/4}(pi/3)^{3/4} 2sqrt{2} rho_0^{3/4}$, a value hydrodynamics cannot reach on its own.
- Stefan-Boltzmann's law with its precise coefficient, $rho = (3/8)pi^2 N_c^2 T^4$, drops out of identifying the Hawking temperature of the dual black hole with the gauge theory temperature, recovering the known strong-coupling result from the time-dependent construction.
- The horizon position depends only on the energy density, so the entire thermodynamic response of the expanding fireball is controlled by one function of proper time.
- The viscosity exponent is constrained to $1/3 <= beta < 5/3$ by horizon regularity of the curvature invariant; below $1/3$ the viscous term dominates and the geometry is singular, above $5/3$ the viscous effect is invisible at the order retained.
Reading between the lines
- Because the construction only requires an equation of state and a value of $eta/s$, the same late-time metric template should transfer to any conformal plasma with a gravity dual, not just $N=4$ SYM; the horizon-to-entropy step is where a non-conformal theory would need modification.
- The result suggests a practical division of labour for fireball modelling: hydrodynamics supplies the evolution of $rho(tau)$, and the dual horizon supplies the absolute normalization of entropy, which is exactly the quantity that elliptic-flow analyses are most sensitive to.
- If the asymptotic entropy is fixed by $rho_0$ alone, then two collisions preparing the same initial energy density but different initial velocity gradients should end at the same $S_infty$; that is a sharp statement, though testing it requires a hydrodynamic code that tracks gradients independently.
- The paper's own extrapolation of $eta propto T^3$ from static to slowly varying backgrounds is the natural place a follow-up calculation would bite: a first-principles retarded-correlator computation in the time-dependent background would either confirm the substitution or produce a correction at the same order as the effect being computed.
Original abstract
We consider a gravity dual description of time dependent, strongly interacting large-Nc N=4 SYM. We regard the gauge theory system as a fluid with shear viscosity. Our fluid is expanding in one direction following the Bjorken's picture that is relevant to RHIC experiments. We obtain the dual geometry at the late time that is consistent with dissipative hydrodynamics. We show that the integration constants that cannot be determined by hydrodynamics are given by looking at the horizon of the dual geometry. Relationship between time dependence of the energy density and bulk singularity is also discussed.
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 a gravity dual description of time dependent, strongly interacting large-Nc N=4 SYM. We regard the gauge theory system as a fluid with shear viscosity. Our fluid is expanding in one direction following the Bjorken's picture that is relevant to RHIC experiments. We obtain the dual geometry at the late time that is consistent with dissipative hydrodynamics. We show that the integration constants that cannot be determined by hydrodynamics are given by looking at the horizon of the dual geometry. Relationship between time dependence of the energy density and bulk singularity is also discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider a gravity dual description of time dependent, strongly interacting large-Nc N=4 SYM. | contribution 0.77 | 0.73 | 0.06 | asserted 0.73 |
| s1 | We regard the gauge theory system as a fluid with shear viscosity. | method 0.64 | 0.50 | 0.06 | asserted 0.99 |
| s2 | Our fluid is expanding in one direction following the Bjorken's picture that is relevant to RHIC experiments. | method 0.50 | 0.41 | 0.05 | asserted 1.00 |
| s3 | We obtain the dual geometry at the late time that is consistent with dissipative hydrodynamics. | result 0.91 | 0.94 | 0.42 | hedged possibility 0.62 |
| s4 | We show that the integration constants that cannot be determined by hydrodynamics are given by looking at the horizon of the dual geometry. | result 0.91 | 0.93 | 0.33 | asserted 1.00 |
| s5 | Relationship between time dependence of the energy density and bulk singularity is also discussed. | contribution 0.53 | 0.74 | 0.08 | asserted 0.68 |
What was examined, and what was not
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References
How the literature uses it
53 arXiv papers cite this in 89 sentences
Papers citing it · 49 sentences without a confident label
Builds on it 2 papers
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We constructed a natural extension of the late-time boost-invariant background found in [ref] (and [ref] – [ref] ) to a background dual to an expanding $=4$ plasma with chemical potential.
Holographic dual of a boost-invariant plasma with chemical potential · Conclusions · 1.00
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Our work builds on earlier derivations of linearized fluid dynamics from linearized gravity by Policastro, Son and Starinets [ref] and on earlier examples of the duality between nonlinear fluid dynamics and gravity by Janik, some of the current authors and collaborators [ref] (, [ref] for related work and extensions and [ref] for some recent work).
Nonlinear Fluid Dynamics from Gravity · Introduction · 0.93
Uses its method or tool 4 papers
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We rely on the pioneering work of Janik and Peschanski [ref] (and important further developments in [ref] ) which sets up a to study boost invariant expansion of a gauge theory plasma in a dual string theory setting.
Second order hydrodynamics of a CFT plasma from boost invariant expansion · Introduction and summary · 0.96
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The introduction of the $r$ dependence of the metric ([ref]) makes solving the Einstein equation a formidable task even with the method of Janik and Peschanski [ref] .
Shear viscosity in late time of hydrodynamic evolution in AdS/CFT duality · Holographic model in late time · 0.81
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We were able to study this system's time evolution thanks to the work in [ref] where a time dependent geometry describing a growing/shrinking black hole in AdS was computed.
Holographic Description of the QCD Phase Diagram and Out of Equilibrium Dynamics · Out of Equilibrium Dynamics · 0.90
Compares with it 5 papers
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In leading order the transverse velocity vanishes and we reproduce the results in Ref. [ref] .
Shear viscosity in late time of hydrodynamic evolution in AdS/CFT duality · Holographic model in late time · 0.96
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The correct expression for the entropy density also follows and we obtain the ratio η/s = (d-1)(d-2)/8π 𝒞_1 (2μ)^1/(d-1) There is no constraint on this ratio at this order because the truncated metric ([ref]) is regular in the bulk [ref] .
Dissipative Bjorken hydrodynamics from an AdS Schwarzschild black hole · Next-to-leading order · 0.88
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Comparing also the bound ([ref]) with that in Eddington-Finkelstein coordinates given by ([ref]), we find some relation between $ε_0$ and $m_0$ using ([ref]): [equation] For $q_0=0$ this relation can be easily checked by our solution with the general form of the metric in [ref] .
Holographic dual of a boost-invariant plasma with chemical potential · Fefferman-Graham vs. Eddington-Finkelstein coordinates · 1.00
Cites it as background 25 papers
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Recent work by Janik and collaborators [ref] (see also [ref] ) opened a possibility to study non-stationary/non-equilibrium processed in gauge theories using their dual string theory formulation [ref] .
Shear viscosity of boost invariant plasma at finite coupling · Introduction · 0.88
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Only recently some progress was made in the studies of strongly coupled nonlinear gauge theory dynamics using string theory methods [ref] .
Consistent holographic description of boost-invariant plasma · Introduction · 0.93
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The analyses have been done to the zeroth order[We call the leading order “zeroth order.” Then, our first-order (second-order) geometry describes the first-order (second-order) dissipative hydrodynamics.] in the expansion [ref] , to the first order [ref] , to the second order [ref] , and to the third order [ref] .
A Holographic Dual of Bjorken Flow · Introduction · 0.93
Discussion
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