Hyperscaling violation, quasinormal modes and shear diffusion

We study quasinormal modes of shear gravitational perturbations for hyperscaling violating Lifshitz theories, with Lifshitz and hyperscaling violating exponents $z$ and $\theta$. The lowest quasinormal mode frequency yields a shear diffusion constant which is in agreement with that obtained in previous work by other methods. In particular for theories with $z< d_i+2-\theta$ where $d_i$ is the boundary spatial dimension, the shear diffusion constant exhibits power-law scaling with temperature, while for $z=d_i+2-\theta$, it exhibits logarithmic scaling. We then calculate certain 2-point functions of the dual energy-momentum tensor holographically for $z\leq d_i+2-\theta$, identifying the diffusive poles with the quasinormal modes above. This reveals universal behaviour $\eta/s=1/4\pi$ for the viscosity-to-entropy-density ratio for all $z\leq d_i+2-\theta$.