Entropy production, viscosity bounds and bumpy black holes

A bstract The ratio of shear viscosity to entropy density, η/s , is computed in various holographic geometries that break translation invariance (but are isotropic). The shear viscosity does not have a hydrodynamic interpretation in such backgrounds, but does quantify the rate of entropy production due to a strain. Fluctuations of the metric components δg xy are massive about these backgrounds, leading to η/s < 1 / (4 π ) at all finite temperatures (even in Einstein gravity). As the temperature is taken to zero, different behaviors are possible. If translation symmetry breaking is irrelevant in the far IR, then η/s tends to a constant at T = 0. This constant can be parametrically small. If the translation symmetry is broken in the far IR (which nonetheless develops emergent scale invariance), then η/s ∼ T 2 ν as T → 0, with ν ≤ 1 in all cases we have considered. While these results violate simple bounds on η/s , we note that they are consistent with a possible bound on the rate of entropy production due to strain.