  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.
