Comments on the holographic shear viscosity to entropy density ratio

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}.