Viscous plasma evolution from gravity using AdS/CFT

Romuald A. Janik

REVIEW

Viscous plasma evolution from gravity using AdS/CFT

Romuald A. Janik

High Energy Physics - Theory · Submitted 12 Oct 2006 · 17 references

Pith's verdict: A landmark dynamical derivation of eta/s = 1/(4pi) that holds up, with one presentational gap a referee should press on.

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arXiv IDhep-th/0610144 (version 2)

Pith IDTH2CGM54, Pith's permanent record of this paper and its review

CitationRomuald A. Janik. "Viscous plasma evolution from gravity using AdS/CFT." arXiv:hep-th/0610144 (2006). https://pith.science/paper/TH2CGM54

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Editor's note

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Short version: this is the paper that turned the nonsingularity criterion into a real dynamical derivation of eta/s = 1/(4pi), and it holds up. The new content is the second-order metric solve (Eq. 20) and the observation that the curvature invariant at order tau^{-4/3} is finite only for eta0^2 = sqrt(3)/18 (Eq. 22), which is exactly the static Policastro-Son-Starinets value transported to the expanding temperature profile (Eq. 25). That match is not a fit; it is a nontrivial consistency check of the correspondence in the nonlinear regime, and it is why this paper is cited throughout the hydrodynamization literature. What it does well: the Einstein equations are actually solved order by order with explicit integrals, the arithmetic connecting Eq. 22 to Eq. 25 checks out, and the author is upfront that C is undetermined at this order and that a next-order analysis is needed. Soft spots, in proportion. The one that matters is presentational: Eq. 21 gives R^2 as a polynomial over (3-v^4)^4(3+v^4)^6 but never displays the polynomial, so the cancellation at v=3^{1/4} is asserted, not shown, and a referee cannot verify the central algebraic step from the text. The stress-test pushes this into 'the system is overdetermined, so generically there is no solution.' I do not buy that stronger claim: the numerator is linear in eta0^2 and C, and the paper's own statement that C drops out is precisely the special factorized structure that lets a single tuning of eta0^2 work. The downstream Heller-Janik-Peschanski work confirmed the value, so the factorization is there. Still, a one-line display of the polynomial would have closed the gap. Second, smaller: the abstract's 'without any further assumptions' overstates things. The hydrodynamic form of epsilon(tau) with free eta0 is an input; what is derived is the coefficient, not the expansion itself. The body is more careful than the abstract. Who it is for: anyone working on holographic hydrodynamics, thermalization, or the eta/s program. It is a foundational reference, not a self-contained derivation. Recommendation: send it to peer review without hesitation; the result is important and correct. Ask for the polynomial in Eq. 21 to be displayed and for the abstract to be softened.

Referee report

Major objections (3)

[Eq. (21) and the paragraph preceding it] The central step of the paper is the statement that R^2 at order tau^{-4/3} is finite exactly when eta0^2 = sqrt(3)/18. Eq. (21) gives this quantity as a polynomial in v, eta0 and C divided by (3-v^4)^4(3+v^4)^6, but the polynomial itself is not displayed. The denominator has a fourth-order pole at v = 3^{1/4}, so finiteness requires the numerator to vanish to fourth order there. Since the second-order coefficients in Eq. (20) are linear in eta0^2 and C, the numerator is linear in these two parameters; with C unrestricted at this order, the C-dependent part must already vanish to fourth order, leaving four conditions on the single parameter eta0^2. Generically this is overdetermined, and a solution exists only if the numerator has a special factorized structure. The paper asserts the cancellation but does not display the polynomial or the factorization, so the reader cannot verify the fourth-order vanishing without redoing the full R^2 computation. Please display the polynomial in Eq. (21), or at least state the factorization that makes the cancellation manifest, and show explicitly that the C-dependent part vanishes to fourth order at v = 3^{1/4}. This is load-bearing for the central claim.
[Abstract and Discussion] The abstract states that nonsingularity 'predicts ... hydrodynamic expansion of the plasma with viscosity coefficient exactly matching' the static result, and the Discussion repeats that the requirement 'predicts the proper-time evolution of the energy density.' In fact the hydrodynamic form of epsilon(tau) in Eq. (5), with eta0 as a free coefficient, is assumed as an ansatz; what the paper derives is the value of that coefficient, not the hydrodynamic expansion itself. The leading perfect-fluid behavior (s = 4/3) is derived from nonsingularity in [8], but the viscous correction is an input. Please soften the framing or state explicitly what is derived versus assumed, so that the abstract's 'without any further assumptions about gauge theory dynamics' is not read as claiming more than the calculation supports.
[Eq. (20) and the discussion of C] The paper acknowledges that C is not fixed at this order and expects the next order to determine it. That acknowledgment is appropriate. What remains unresolved is the claim, implicit in the cancellation argument, that the C-dependent part of R^2 vanishes to fourth order at v = 3^{1/4} so that the eta0 extraction is well defined independently of C. This is a nontrivial structural fact about the explicit coefficients in Eq. (20) and should be demonstrated, not merely asserted. If the C-part does not vanish to fourth order, the cancellation condition would involve C and the extraction of eta0 would not be unique at this order.

Minor objections (5)

[Eq. (16)] The first-order coefficients a1, b1, c1 are taken from Nakamura and Sin [10]. Please state this explicitly at the equation and confirm that the sign and normalization conventions match those of [10], since the second-order result inherits any convention mismatch.
[Eq. (20)] The integrals defining a2, b2, c2 are left unevaluated, with the remark that they can be expressed in terms of elementary functions and dilogarithms. For a letter this is acceptable, but the specific combination of these integrals that enters R^2 at order tau^{-4/3} should be given, since that is the object whose vanishing is the crux of the paper.
[Notation] The paper uses both eta and eta0; please state once, near Eq. (4), that eta0 is the coefficient in eta = eta0/tau, to avoid ambiguity with the temperature-dependent eta of Eq. (23).
[Eq. (24)] The displayed factors in Eq. (24) are difficult to parse as typeset. Please ensure the numerical prefactors are clearly separated so that the reader can verify the arithmetic connecting Eq. (23) to Eq. (25).
[Eq. (14)] The factor sqrt(2) in the temperature is attributed to the Fefferman-Graham form of the black-hole metric and referred to [8]. A one-sentence reminder of its origin would help the reader who has not worked through [8].

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The paper

The paper claims

Janik argues that demanding the five-dimensional dual geometry of an expanding plasma stay free of curvature singularities forces the plasma's shear viscosity to a single value, the same one found earlier for a static plasma.

The reading

The paper asks whether the dynamics of a strongly coupled fluid can be extracted from the geometry that duals it, without assuming anything about the gauge theory's microscopic dynamics. Janik studies a boost-invariant expanding fireball of $N=4$ super Yang-Mills theory, builds its five-dimensional dual metric order by order in a late-time expansion, and demands that the curvature invariant $R^2$ stay finite everywhere. At second subleading order the metric coefficients carry the viscosity coefficient $eta_0$ as a free parameter, and the would-be singularity at $v = 3^{1/4}$ cancels for exactly one value of it. That value agrees with what one gets by taking the static-plasma result $eta/s = 1/(4pi)$ and inserting the temperature of the expanding plasma, with no fitting. If the argument holds, a purely gravitational smoothness condition reproduces a transport coefficient of a strongly coupled quantum fluid at strong coupling, which is why the match is worth taking seriously.

What carries the argument

The load-bearing device is the nonsingularity criterion applied term by term in a late-time expansion of the dual metric. Fefferman-Graham coordinates put the metric in the form $(tilde{g}_{mu nu}dx^mu dx^nu + dz^2)/z^2$ with $tilde{g}_{mu nu} = eta_{mu nu} + z^4 tilde{g}^{(4)}_{mu nu} + dots$, and $tilde{g}^{(4)}$ is read as the gauge theory's energy-momentum tensor. The scaling variable $v = z tau^{-1/3}$ turns the Einstein equations into ordinary differential equations in $v$, ordered by powers of $tau^{-2/3}$; the leading order reproduces the perfect-fluid geometry with a horizon at $z_0 = 3^{1/4} tau^{1/3}$, the first subleading order reproduces the $O(tau^{-2/3})$ corrections already published elsewhere, and the second order is where the viscosity coefficient first becomes visible in the curvature invariant. The denominator $(3-v^4)^4(3+v^4)^6$ of the $O(tau^{-4/3})$ term of $R^2$ is what localizes the test to a single point in $v$.

Extended reading notes

Core claim

The claim is that nonsingularity of the dual geometry, applied at subasymptotic times, selects viscous hydrodynamics with a uniquely fixed shear viscosity. Concretely: writing the metric as $ds^2 = (-e^{a}d tau^2 + tau^2 e^{b}dy^2 + e^{c}dx_perp^2 + dz^2)/z^2$ and expanding $a = a(v) + a_1(v) tau^{-2/3} + a_2(v) tau^{-4/3} + dots$ in the scaling variable $v = z tau^{-1/3}$, Janik solves the second-order Einstein equations $E^{(2)}_A = 0$ with coefficients that vanish at $v=0$. The resulting curvature invariant, expanded to order $tau^{-4/3}$, takes the form $R^2 = text{nonsingular} + tau^{-4/3} times text{a polynomial in } v, eta_0, C text{ divided by } (3-v^4)^4(3+v^4)^6$, so the only candidate singularity sits at $v = 3^{1/4}$. It cancels exactly when $eta_0^2 = sqrt{3}/18$, and this is precisely the value $eta_0 = (sqrt{3}/18)^{1/2}$ obtained by transporting $eta = s/(4pi)$ with $s = pi^2 N_c^2 T^3/2$ to the expanding temperature $T = sqrt{2} pi^{-1} 3^{-1/4} tau^{-1/3}$. The author notes that the second coefficient $C$ is not fixed at this order and expects a further order in $tau^{-2/3}$ to determine it.

Load-bearing premise

The argument rests on the premise that requiring the curvature invariant to stay finite order by order in the late-time $tau^{-2/3}$ expansion is the correct physical selector, so that a singularity appearing at order $tau^{-4/3}$ must be cancelled rather than being an artifact of truncating the expansion.

Editorial extensions

If this is right

The shear viscosity of the strongly coupled plasma is fixed by a regularity condition on a nonlinear, time-dependent gravity solution, not by a linear-response calculation around a static black hole.
The late-time expansion of the energy density takes the hydrodynamic form $varepsilon(tau) = tau^{-4/3} - 2eta_0 tau^{-2} + dots$ with $eta_0 = (sqrt{3}/18)^{1/2}$, so the approach to perfect-fluid behaviour is quantitatively predicted.
The same nonsingularity method becomes a candidate tool for extracting transport coefficients in other time-dependent, non-equilibrium processes where linear response is unavailable.
The result supplies an independent consistency check of the AdS/CFT correspondence at the level of a specific number, since the dynamical and static routes to $eta_0$ agree.

Reading between the lines

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

If the criterion is trustworthy, the same machinery should give the second-order hydrodynamic transport coefficients (the relaxation time and the curvature couplings) once the calculation is pushed one order further, since the unfixed constant $C$ is presumably where that information sits.
The match is a check of a proposed selection rule rather than of the correspondence itself: a different but equally plausible regularity condition at each order could in principle have selected a different $eta_0$, so the agreement is evidence that the order-by-order criterion is the physically correct one.
Because the argument never uses the microscopic Lagrangian, it suggests that any gravitational dual with a smooth horizon responds to expansion in the same hydrodynamic way, which would make $eta/s = 1/(4pi)$ a consequence of horizon regularity rather than of supersymmetry.

Original abstract

We analyze the AdS/CFT dual geometry of an expanding boost-invariant plasma. We show that the requirement of nonsingularity of the dual geometry for leading and subasymptotic times predicts, without any further assumptions about gauge theory dynamics, hydrodynamic expansion of the plasma with viscosity coefficient exactly matching the one obtained earlier in the static case by Policastro, Son and Starinets.

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The abstract, sentence by sentence

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We analyze the AdS/CFT dual geometry of an expanding boost-invariant plasma. We show that the requirement of nonsingularity of the dual geometry for leading and subasymptotic times predicts, without any further assumptions about gauge theory dynamics, hydrodynamic expansion of the plasma with viscosity coefficient exactly matching the one obtained earlier in the static case by Policastro, Son and Starinets.

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s0	We analyze the AdS/CFT dual geometry of an expanding boost-invariant plasma.	contribution 0.92	0.89	0.04	asserted 0.94
s1	We show that the requirement of nonsingularity of the dual geometry for leading and subasymptotic times predicts, without any further assumptions about gauge theory dynamics, hydrodynamic expansion of the plasma with viscosity coefficient exactly matching the one obtained earlier in the static case by Policastro, Son and Starinets.	result 1.00	0.91	0.91	asserted 1.00

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References

Works cited (17)

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[4]

G. Policastro, D. T. Son and A. O. Starinets, “The shear vi scosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma,” Phys. Rev. Lett. 87, 081601 (2001) [arXiv:hep-th/0104066]

View on Pith Pith review arXiv 2001
[5]

G. Policastro, D. T. Son and A. O. Starinets, “From AdS/CF T correspondence to hydrodynamics,” JHEP 0209, 043 (2002) [arXiv:hep-th/0205052]; P. Kovtun, D. T. Son and A. O. Starinets, “Holography and hydr odynamics: Diﬀusion on stretched horizons,” JHEP 0310, 064 (2003) [arXiv:hep-th/0309213]; P. Kovtun, D. T. Son and A. O. Starinets, “Viscosity in strong ...

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[6]

H. Nastase, “The RHIC ﬁreball as a dual black hole,” arXiv :hep-th/0501068

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[7]

E. Shuryak, S. J. Sin and I. Zahed, “A gravity dual of RHIC c ollisions,” arXiv:hep-th/0511199

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[9]

R. A. Janik and R. Peschanski, “Gauge / gravity duality an d thermalization of a boost-invariant perfect ﬂuid,” Phys. Rev. D 74 (2006) 046007 [arXiv:hep-th/0606149]

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J. D. Bjorken, “Highly Relativistic Nucleus-Nucleus C ollisions: The Central Rapidity Region,” Phys. Rev. D 27, 140 (1983)

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[12]

S. de Haro, S. N. Solodukhin and K. Skenderis, “Holograp hic reconstruction of spacetime and renormalization in the AdS/CFT correspondence,” Commun. Math. Phys. 217, 595 (2001) [arXiv:hep-th/0002230]; K. Skenderis, “Lecture notes on holographic renormalizati on,” Class. Quant. Grav. 19, 5849 (2002) [arXiv:hep-th/0209067]

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[14]

M. Heller, R.A. Janik, R. Peschanski, work in progress

View on Pith
[15]

In that paper f (τ ) ≡ ε(τ )

View on Pith
[16]

The power of τ represents η ∝ T 3

View on Pith
[17]

Up to the considered order in 1 /τ

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50 later papers cite it

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1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 6 papers with this label

Rescaled the Einstein equations as in this paper to obtain a uniform expansion.

Consistent holographic description of boost-invariant plasma (2008) · 0.93 · paraphrased: the licence does not allow quoting

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49 arXiv papers cite this in 115 sentences

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Papers citing it · 72 sentences without a confident label

Contests it 1 paper

First, the nonsingularity approach of [ref] is simply inconsistent — it was shown in [ref] that it is definitely inconsistent within the supergravity approximation at high orders in the late proper time expansion ( though it appears to be well-defined in the first two orders necessary to compute shear viscosity of $=4$ plasma in the supergravity approximation ).

Shear viscosity of boost invariant plasma at finite coupling · Conclusion · contest 0.84

Builds on it 1 paper

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 6 papers

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
For the five-dimensional metric we use the same ansatz as in [ref] : [equation] where $dx_⊥^2≡dx_1^2+dx_2^2$.

On the supergravity description of boost invariant conformal plasma at strong coupling · Equations of motion · 0.93
In principle, analysis are very straightforward: we compute $'$ corrections to the gravitational background dual to boost invariant Bjorken expansion on the boundary; as in [ref] , we fix some of the parameters[Amusingly, not all parameters can be fixed from the nonsingularity condition [ref] .

Shear viscosity of boost invariant plasma at finite coupling · Shear viscosity of the $=4$ SYM plasma at finite 't Hooft coupling · 0.80

Compares with it 6 papers

The difference here (compare to [ref] ) is that $_0$ and $C_0$ are already fixed by requiring the nonsingularity of the Ricci scalar.

Second order hydrodynamics of a CFT plasma from boost invariant expansion · $$ at order $(^-2)$ · 0.82
The difference here (compare to [ref] ) is that $_0$ is already fixed by requiring the nonsingularity of the Ricci scalar[Notice that Ricci scalar vanishes in the supergravity approximation [ref] .].

Shear viscosity of boost invariant plasma at finite coupling · $$ · 1.00
The perturbed metric was shown to be non-singular in the bulk, provided η/s = 1/4π where $s$ is the entropy density, in agreement with asymptotic time-dependent solutions in five dimensions [ref] .

Black holes and the quark-gluon plasma · Introduction · 0.98

Cites it as background 25 papers

Supergravity dual to boost invariant expanding $=4$ QGP was discussed in [ref] .

On the supergravity description of boost invariant conformal plasma at strong coupling · $=4$ QGP · 0.99
Janik and Peschanski first considered a proper time dependent $AdS_5$ metric with boost invariance through holographic renormalization in late time of fluid evolution [ref] .

Shear viscosity in late time of hydrodynamic evolution in AdS/CFT duality · Introduction · 0.89
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

Discussion

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M. Heller, R.A. Janik, R. Peschanski, work in progress

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In that paper f (τ ) ≡ ε(τ )

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The power of τ represents η ∝ T 3

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[17] [17]

Up to the considered order in 1 /τ

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