The Viscosity Bound Conjecture and Hydrodynamics of M2-Brane Theory at Finite Chemical Potential

TL;DR

This paper investigates the viscosity to entropy density ratio in M2-brane theory at finite chemical potential, finding that the bound is saturated even with non-zero R-charge density, supporting the universality of the viscosity bound.

Contribution

It demonstrates that the shear viscosity to entropy density ratio remains at the bound up to fourth order corrections in a finite chemical potential background for M2-branes.

Findings

Viscosity increases with R-charge density.

Corrections to η/s vanish up to fourth order.

Bound remains saturated with finite chemical potential.

Abstract

Kovtun, Son and Starinets have conjectured that the viscosity to entropy density ratio $\eta/s$ is always bounded from below by a universal multiple of $\hbar$ i.e., $\hbar/(4\pi k_{B})$ for all forms of matter. Mysteriously, the proposed viscosity bound appears to be saturated in all computations done whenever a supergravity dual is available. We consider the near horizon limit of a stack of M2-branes in the grand canonical ensemble at finite R-charge densities, corresponding to non-zero angular momentum in the bulk. The corresponding four-dimensional R-charged black hole in Anti-de Sitter space provides a holographic dual in which various transport coefficients can be calculated. We find that the shear viscosity increases as soon as a background R-charge density is turned on. We numerically compute the few first corrections to the shear viscosity to entropy density ratio $\eta/s$ and surprisingly discover that up to fourth order all corrections originating from a non-zero chemical potential vanish, leaving the bound saturated. This is a sharp signal in favor of the saturation of the viscosity bound for event horizons even in the presence of some finite background field strength. We discuss implications of this observation for the conjectured bound.