Viscosity of $R^2$ Modified AdS Black Brane

We investigate the Einstein-Hilbert black brane solution in four-dimensional Anti-de Sitter (AdS) spacetime supplemented by a quadratic Ricci scalar term $q L^2 R^2$, where $q$ is a dimensionless coupling constant and $L$ is the AdS radius. The shear viscosity to entropy density ratio, $\frac{\eta}{s}$, is calculated holographically, and deviations from the universal Kovtun-Son-Starinets (KSS) bound are analyzed. Our results indicate that $\frac{\eta}{s} = \frac{1}{4\pi}(1 - 24q)$, demonstrating that the ratio falls below the conjectured lower limit for positive $q$, while it respects the bound for negative $q$. We confirm that our solutions smoothly reduce to the standard Einstein-Hilbert case when $q \to 0$, consistent with expectations. The physical implications of violating the KSS bound are discussed in depth, particularly regarding stability, causality, and the strongly coupled nature of the dual field theory. These findings provide valuable insights into the influence of higher curvature terms on holographic transport properties.