Nonlinear Fluid Dynamics from Gravity (0712.2456v4)

Abstract: Black branes in AdS5 appear in a four parameter family labeled by their velocity and temperature. Promoting these parameters to Goldstone modes or collective coordinate fields -- arbitrary functions of the coordinates on the boundary of AdS5 -- we use Einstein's equations together with regularity requirements and boundary conditions to determine their dynamics. The resultant equations turn out to be those of boundary fluid dynamics, with specific values for fluid parameters. Our analysis is perturbative in the boundary derivative expansion but is valid for arbitrary amplitudes. Our work may be regarded as a derivation of the nonlinear equations of boundary fluid dynamics from gravity. As a concrete application we find an explicit expression for the expansion of this fluid stress tensor including terms up to second order in the derivative expansion.

• The paper derives nonlinear fluid dynamics from gravitational systems using a systematic perturbative expansion of black brane dynamics.
• It computes a boundary stress tensor to second order, confirming the universal shear viscosity relation η/s = 1/(4π) in conformal fluids.
• The work bridges microscopic quantum field theories with macroscopic fluid behavior, offering insights applicable to heavy-ion collision experiments.

Nonlinear Fluid Dynamics from Gravity

The academic paper at hand explores the intriguing relationship between nonlinear fluid dynamics and gravity within the context of the AdS/CFT correspondence, a pivotal framework in theoretical physics. The authors, Bhattacharyya, Hubeny, Minwalla, and Rangamani, formulate a systematic procedure to derive fluid dynamics from the dynamics of gravitational systems in Anti-de Sitter (AdS) spaces. This approach effectively bridges the gap between microscopic quantum field theories and macroscopic phenomena like fluid behavior.

Main Contributions

The key contribution of this paper is the derivation of nonlinear equations of boundary fluid dynamics from gravitational theories. Specifically, the authors formulate a boundary stress tensor, expanded to the second order in the derivative expansion, by promoting the dynamics of black brane parameters (velocity and temperature) to collective Goldstone modes. This derived stress tensor, reflecting the fluid's properties, conforms to the expectations from gravitational dynamics as predicted by the AdS/CFT duality.

Theoretical Framework and Methodology

The work builds upon earlier studies of linearized fluid dynamics from gravity and extends these to nonlinear regimes. The authors utilize black brane solutions in AdS spaces as laboratories to explore the equivalent dynamics of strongly coupled conformal field theories at long wavelengths. The methodology involves using Einstein's equations, under specific regularity conditions and boundary constraints, to map out equations of boundary fluid dynamics.

The paper meticulously outlines the perturbative expansion approach, where fluid parameters like velocity and temperature are treated as functions of boundary coordinates. In this framework, the perturbative solution of gravitational equations hinges on a boundary derivative expansion, leading to a universal characterization of fluid dynamics in the dual field theory context.

Significant Results and Implications

A pivotal result articulated by the authors is the computation of a boundary fluid stress tensor, which incorporates terms up to the second order in the derivative expansion. The derived shear viscosity coefficient confirms the known relation η/s = 1/(4π) for all conformal fluids with gravitational duals, underscoring a certain universality in these descriptions.

The implications of these findings are substantial, both theoretically and practically. From a theoretical standpoint, the work offers a robust framework for understanding the fluid dynamics of strongly coupled field theories, validating fluid dynamics as an appropriate long-wavelength description for these systems. Practically, this correspondence has relevance in analyzing real-world phenomena such as heavy-ion collision experiments, reflected in comparisons with RHIC data.

Prospects for Future Research

The research opens several avenues for further exploration. Pertinent among these is the potential application to broader classes of gravitational theories interacting with fields beyond pure gravity, such as incorporating gauge fields. This could lead to nuanced descriptions of charged fluid dynamics in theories with non-trivial string theory backgrounds.

Additionally, the paper hints at potential insights into gravitational phenomena like cosmic censorship through these fluid dynamics mappings. Exploring the existence and nature of event horizons in these derived solutions offers intriguing prospects for advancing our understanding of black hole physics.

Furthermore, the investigation of the universality of higher-order transport coefficients, alongside studies of their sensitivity to corrections from higher derivative terms in the gravitational action, constitutes a fertile ground for future theoretical insights.

Overall, this paper provides a foundational piece in understanding the intricate dance between gravity and fluid dynamics, contributing significantly to the broader dialogue surrounding the implications of the AdS/CFT correspondence.