Comments on the holographic shear viscosity to entropy density ratio

TL;DR

This paper clarifies that the shear viscosity in holographic models is a fourth-rank tensor and derives three distinct formulas for its ratio to entropy density, revealing universal and non-universal behaviors depending on tensor components.

Contribution

It highlights the tensorial nature of shear viscosity in holography and derives new formulas showing different universalities for various tensor components.

Findings

a^j_i^j_i/s ratio is universally 1/(4\u03c0)

a^{jiji}/s and a_{jiji}/s ratios depend on background metric details

Different tensor components yield distinct shear viscosity to entropy density ratios.

Abstract

We revisit the membrane paradigm calculations of the holographic shear viscosity tensor of strongly coupled isotropic plasmas with Einstein gravity dual by emphasizing the fact which was overlooked in the previous literatures that the shear viscosity is a fourth-rank tensor. Using the membrane paradigm we show that depending on whether the holographic shear viscosity tensor to entropy density ratio is \eta^j_i^j_i/s or \eta^{jiji}/s or \eta_{jiji}/s, we can derive three distinct formulae for the holographic shear viscosity tensor to entropy density ratios given explicitly in terms of the background metric g_{ij}. We find that the widely studied \eta^j_i^j_i/s holographic shear viscosity tensor to entropy density ratio takes the universal value 1/(4\pi) for isotropic background metric g_{ij} but \eta^{jiji}/s and \eta_{jiji}/s holographic shear viscosity tensor to entropy density ratios take non-universal values which depend on the details of the isotropic background metric g_{ij}.