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Relativistic viscous hydrodynamics, conformal invariance, and holography

R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, M. A. Stephanov

TL;DR

The paper develops a complete second-order, conformally invariant relativistic hydrodynamics framework, showing that conformal symmetry reduces the allowed second-order terms to five. Using AdS/CFT, it determines three coefficients (τ_Π, κ, λ_1) for strongly coupled N=4 SYM, while kinetic theory at weak coupling indicates some coefficients vanish and κ can be nonzero, illustrating the limits of kinetic derivations. It also critiques the Müller-Israel-Stewart approach for missing conformally required terms and highlights how higher-order (non-hydrodynamic) modes arise, requiring an extended set of variables for full gravity-like descriptions. The results have direct implications for modeling the quark-gluon plasma and for accurate numerical simulations of conformal or near-conformal fluids. Overall, the work provides a rigorous bridge between microscopic theory, holography, and macroscopic hydrodynamics in the conformal regime, with practical guidance for QGP phenomenology.

Abstract

We consider second-order viscous hydrodynamics in conformal field theories at finite temperature. We show that conformal invariance imposes powerful constraints on the form of the second-order corrections. By matching to the AdS/CFT calculations of correlators, and to recent results for Bjorken flow obtained by Heller and Janik, we find three (out of five) second-order transport coefficients in the strongly coupled N=4 supersymmetric Yang-Mills theory. We also discuss how these new coefficents can arise within the kinetic theory of weakly coupled conformal plasmas. We point out that the Mueller-Israel-Stewart theory, often used in numerical simulations, does not contain all allowed second-order terms and, frequently, terms required by conformal invariance.

Relativistic viscous hydrodynamics, conformal invariance, and holography

TL;DR

The paper develops a complete second-order, conformally invariant relativistic hydrodynamics framework, showing that conformal symmetry reduces the allowed second-order terms to five. Using AdS/CFT, it determines three coefficients (τ_Π, κ, λ_1) for strongly coupled N=4 SYM, while kinetic theory at weak coupling indicates some coefficients vanish and κ can be nonzero, illustrating the limits of kinetic derivations. It also critiques the Müller-Israel-Stewart approach for missing conformally required terms and highlights how higher-order (non-hydrodynamic) modes arise, requiring an extended set of variables for full gravity-like descriptions. The results have direct implications for modeling the quark-gluon plasma and for accurate numerical simulations of conformal or near-conformal fluids. Overall, the work provides a rigorous bridge between microscopic theory, holography, and macroscopic hydrodynamics in the conformal regime, with practical guidance for QGP phenomenology.

Abstract

We consider second-order viscous hydrodynamics in conformal field theories at finite temperature. We show that conformal invariance imposes powerful constraints on the form of the second-order corrections. By matching to the AdS/CFT calculations of correlators, and to recent results for Bjorken flow obtained by Heller and Janik, we find three (out of five) second-order transport coefficients in the strongly coupled N=4 supersymmetric Yang-Mills theory. We also discuss how these new coefficents can arise within the kinetic theory of weakly coupled conformal plasmas. We point out that the Mueller-Israel-Stewart theory, often used in numerical simulations, does not contain all allowed second-order terms and, frequently, terms required by conformal invariance.

Paper Structure

This paper contains 28 sections, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conformal invariance in hydrodynamics
  3. Conformal invariance and Weyl anomalies
  4. First order hydrodynamics as derivative expansion
  5. Conformal invariance in first-order hydrodynamics
  6. Second-order hydrodynamics of a conformal fluid
  7. Second-order terms
  8. Kubo's formulas
  9. Sound Pole
  10. Shear pole
  11. Bjorken Flow
  12. Second-order hydrodynamics for strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills plasma
  13. Scalar channel
  14. Shear channel
  15. Sound channel
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Sound dispersion $c_s=c_s(\textswab{q})$ in ${\cal N}=4$ SYM plasma. The dark (blue) curve shows the sound speed dependence on wavevector, $c_s(\textswab{q}) = \hbox{Re}\, \textswab{w}/ \textswab{q}$, with $c_s(0)=1/\sqrt{3}$ (this plot is based on numerical data first obtained in Kovtun:2005ev). The light (red) curve corresponds to analytic approximation derived from Eq. (\ref{['sound_d_rel']}) and valid for sufficiently small $\textswab{q}$.