Linearized fluid/gravity correspondence: from shear viscosity to all order hydrodynamics

TL;DR

This paper develops a comprehensive framework for all-order linearized hydrodynamics in strongly coupled gauge theories using holography, enabling the resummation of derivative terms and detailed analysis of transport coefficients.

Contribution

It provides explicit RG flow equations for transport coefficients and demonstrates how generalized Navier-Stokes equations emerge from holographic constraints.

Findings

Derived closed-form RG flow equations for viscosity functions.

Computed spectrum of small fluctuations up to third order.

Numerically solved RG equations to include all derivative orders.

Abstract

In ref. \cite{1406.7222}, we reported a construction of all order linearized fluid dynamics with strongly coupled $\mathcal{N}=4$ super-Yang-Mills theory as underlying microscopic description. The linearized fluid/gravity correspondence makes it possible to resum all order derivative terms in the fluid stress tensor. Dissipative effects are fully encoded by the shear term and a new one, emerging starting from third order in hydrodynamic derivative expansion. In this work, we provide all computational details omitted in \cite{1406.7222} and present additional results. We derive closed-form linear holographic RG flow-type equations for momenta-dependent transport coefficient functions. Generalized Navier-Stokes equations are shown to emerge from the constraint components of the bulk Einstein equations. We perturbatively solve the RG equations for the viscosity functions, up to third order in derivative expansion, and up to this order compute spectrum of small fluctuations. Finally, we solve the RG equations numerically, thus accounting for all order derivative terms in the boundary stress tensor.