Hydrodynamics of cold holographic matter

We show that at any temperature, the low-energy (with respect to the chemical potential) collective excitations of the transverse components of the energy-momentum tensor and the global U(1) current in the field theory dual to the planar RN-AdS4 black hole are simply those of hydrodynamics. That is, hydrodynamics is applicable even at energy scales much greater than the temperature. It is applicable even at zero temperature. Specifically, we find that there is always a diffusion mode with diffusion constant proportional to the ratio of entropy density to energy density. At low temperatures, the leading order momentum and temperature dependences of the dispersion relation of this mode are controlled by the dimension of an operator in the thermal CFT1 dual to the near-horizon Schwarzschild-AdS2 geometry.