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

TL;DR

The paper develops a linearized, all-order hydrodynamic description for a strongly coupled N=4 SYM plasma via the fluid/gravity correspondence. It derives linear holographic RG-flow equations for momentum-dependent transport functions, including $η(ω,q^2)$ and $ζ(ω,q^2)$, and shows generalized Navier–Stokes dynamics emerge from bulk constraint equations. Perturbative analysis yields explicit third-order contributions and dispersion relations for shear and sound modes, while numerical solutions provide all-order viscosity functions and confirm causal behavior by their vanishing at large momenta. The framework offers a consistent, causal extension of relativistic hydrodynamics that resums infinite derivative corrections and illuminates the role of higher-gradient effects in strongly coupled plasmas, with potential applications to quark–gluon plasma modeling and beyond.

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.

Figures (4)

• Figure 1: The viscosities $\eta(\omega,q^2)$ and $\zeta(\omega,q^2)$ as functions of $\omega$ and $q^2$.
• Figure 2: The generalized viscosity functions $\eta$ and $\zeta$ vs $\omega$ at $q^2=0$.
• Figure 3: The generalized viscosity functions $\eta$ and $\zeta$ vs $q^2$ at $\omega=0$.
• Figure 4: The viscosity functions $\eta(\omega,q^2)$ and $\zeta(\omega,q^2)$ vs $\omega$ and $q^2$ from the matching method.