Viscosity for anisotropic Reissner-Nordström black branes

We investigate the behavior of shear viscosity in the presence of small anisotropy and a finite chemical potential. First, we construct an anisotropic Reissner Nordström black brane in five dimensions in a simple Einstein-Maxwell theory with a small linear dilaton. This solution is characterized by three mass scales: anisotropy ρ, temperature T, and chemical potential μ. We find this solution up to second order in the dilaton anisotropy parameter ρ. This black brane solution corresponds to an anisotropic phase where the anisotropy is small compared to the temperature and chemical potential. We find that in this anisotropic phase, some components of the anisotropic shear viscosity tensor, which are spin one with respect to the residual symmetry after breaking rotational invariance, violates the KSS bound (ηs≥14π) proposed by Kovtun, Son, and Starinets. We identify the regions of the parameter space where these violations are significant. We carry out a similar analysis in four dimensions and find a similar violation of the KSS bound for the spin one components to demonstrate the generality of the result. Our results are particularly relevant in the context of strongly coupled systems found in nature. We also provide an intuitive understanding of the results using dimensional reduction and a Boltzmann calculation in a weakly coupled version of a similar system. The Boltzmann analysis performed in a system of weakly interacting particles in a linear potential also shows that components of the viscosity tensor may be reduced. It is intriguing that the Boltzmann analysis also predicts the corrections to be negative and that too in a manner similar to the anisotropic strongly coupled theories with smooth gravity duals.