# Viscosity for Anisotropic Reissner Nordstr\"om Blackbranes

## TL;DR

This paper studies how small anisotropy and finite chemical potential affect shear viscosity in Reissner Nordström black branes, revealing violations of the KSS bound in certain anisotropic conditions.

## Contribution

It constructs an anisotropic black brane solution in 5D Einstein-Maxwell theory and demonstrates that some shear viscosity components violate the KSS bound due to anisotropy.

## Key findings

- Certain shear viscosity components violate the KSS bound in anisotropic phases.
- Violations are significant in specific regions of parameter space.
- Similar violations occur in 4D models, indicating a general phenomenon.

## Abstract

We investigate the behavior of shear viscosity in the presence of small anisotropy and a finite chemical potential. First, we construct an anisotropic Reissner Nordstr{\"o}m blackbrane in 5 dimensions in a simple Einstein-Maxwell theory with a small linear dilaton. This solution is characterized by three mass scales : anisotropy $\rho$, temperature T and chemical potential $\mu$. We find this solution upto second order in the dilaton anisotropy parameter $\rho$. This blackbrane 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 (${\eta \over s}\ge {1 \over 4 \pi} $) 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 4 dimensions and find 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.