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 $\eta/s \sim T^{2 \nu}$ as $T \to 0$, with $\nu \leq 1$ 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.