We compute the strong coupling limit of the shear viscosity for the $\mathcal{N}=4$ super-Yang-Mill theory with a chemical potential. We use the five-dimensional Reissner-Nordström-anti-deSitter black hole, so the chemical potential is the one for the $R$-charges $U(1{)}_R^3$. We compute the quasinormal frequencies of the gravitational and electromagnetic vector perturbations in the background numerically. This enables one to explicitly locate the diffusion pole for the shear viscosity. The ratio of the shear viscosity $\eta$ to the entropy density $s$ is $\eta /s=1/(4\pi )$ within numerical errors, which is the same result as the one without chemical potential.

• Received 2 February 2006

DOI:https://doi.org/10.1103/PhysRevD.73.066013