Entropy production, viscosity bounds and bumpy black holes

The ratio of shear viscosity to entropy density, eta/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 delta g(xy) are massive about these backgrounds, leading to eta/s < 1/(4 pi) 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 eta/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 eta/s similar to T-2 nu as T -> 0, with nu <= 1 in all cases we have considered. While these results violate simple bounds on eta/s, we note that they are consistent with a possible bound on the rate of entropy production due to strain.