Relativistic viscous hydrodynamics, conformal invariance, and holography (0712.2451v3)

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.

• The paper derives second-order hydrodynamic equations for conformal fluids, showing how conformal invariance restricts the allowed transport coefficients.
• The study computes key coefficients such as τΠ = (2 - ln2)/(2πT) and κ = η/(πT), validating predictions with holographic methods.
• The paper contrasts strong coupling results with kinetic theory, emphasizing that terms like κ and λ3 are absent in weakly coupled descriptions.

Overview of "Relativistic Viscous Hydrodynamics, Conformal Invariance, and Holography"

The paper "Relativistic Viscous Hydrodynamics, Conformal Invariance, and Holography" by Baier et al. provides an extensive investigation into the dynamics of conformal fluids through the lens of hydrodynamics, particularly focusing on the second-order effects in strongly coupled systems, which are often observed in quantum field theories such as the ${\cal N}=4$ supersymmetric Yang-Mills (SYM) theory at finite temperature. This paper involves the derivation and analysis of hydrodynamic equations to second-order in gradient expansion under conformal symmetry constraints, aiming to provide theoretical predictions for the behavior of the stress-energy tensor in such systems.

The central result of the paper is the identification and calculation of second-order transport coefficients for the strongly coupled ${\cal N}=4$ SYM plasma using the AdS/CFT correspondence. These coefficients include the shear viscosity $\eta$, relaxation time $\tau_\Pi$, as well as other parameters like $\kappa$, and $\lambda_{1,2,3}$. Here the authors specifically find that conformal invariance imposes significant limitations on the possible form of second-order corrections, effectively reducing the number of independent transport coefficients compared to a generic non-conformal system.

Key Contributions and Results

1. Second-Order Hydrodynamics in Conformal Field Theories: The paper rigorously derives the hydrodynamic equations incorporating second-order derivative corrections for conformal fluids. The authors identify five such corrections and determine that, notably, the M\"uller-Israel-Stewart theory does not encompass all allowable second-order terms demanded by conformal invariance.
2. Computation of Transport Coefficients using Holography: Through matching hydrodynamic expressions to AdS/CFT results for correlators and Bjorken flow characteristics, the paper calculates specific second-order transport coefficients. For ${\cal N}=4$ SYM, $\tau_\Pi$ was found to be $(2 - \ln 2)/(2 \pi T)$, with $\kappa = \eta / (\pi T)$ and $\lambda_1$ proportional to $\eta$. These findings are corroborated using independent calculations within the AdS/CFT framework, contributing valuable insights into the dynamics of strong coupling regimes.
3. Implications for Weakly Coupled Kinetic Theory: By exploring the kinetic theory for weakly coupled conformal plasmas, the analysis shows that not all second-order terms are present within kinetic descriptions, particularly highlighting the absence of $\kappa$ and $\lambda_3$. This suggests the emergence of additional effects in strongly coupled scenarios that cannot be captured through conventional kinetic theory approaches.
4. Extension and Comparisons: The paper outlines how the M\"uller-Israel-Stewart framework can be extended by deriving expressions that remain valid within its phenomenological range, emphasizing the necessity of more complete terms to accurately reflect conformal invariance. This includes extensions for relativistic hydrodynamics simulations, relevant for theoretical studies of systems like the quark-gluon plasma produced in heavy ion collisions.

Implications and Future Directions

The authors' work on second-order hydrodynamics in conformal field theories provides critical insights into the realistic modeling of strongly coupled plasmas like those found in the early universe or created in high-energy particle physics experiments. The detailed calculations offer benchmarks for simulations trying to capture the complex dynamics of these systems more accurately. Future research could further explore the connection between the emergent transport coefficients in strongly coupled theories and their weakly coupled counterparts, potentially refining the bridge between kinetic theory and holographic descriptions. Additionally, investigating the precise nature of terms absent in kinetic theories at weak coupling might illuminate new aspects of thermal field theories and their effective descriptions through holography.

Overall, this paper sets a foundational platform for researchers aiming to explore the dynamics of complex quantum field theories and aids in enhancing the understanding of transport processes in strongly interacting systems.