Momentum relaxation in holographic massive gravity

We study the effects of momentum relaxation on observables in a recently proposed holographic model in which the conservation of momentum in the field theory is broken by the presence of a bulk graviton mass. In the hydrodynamic limit, we show that these effects can be incorporated by a simple modification of the energy-momentum conservation equation to account for the dissipation of momentum over a single characteristic time scale. We compute this time scale as a function of the graviton mass terms and identify the previously known "wall of stability" as the point at which this relaxation time scale becomes negative. We also calculate analytically the zero temperature AC conductivity at low frequencies. In the limit of a small graviton mass this reduces to the simple Drude form, and we compute the corrections to this which are important for larger masses. Finally, we undertake a preliminary investigation of the stability of the zero temperature black brane solution of this model, and rule out spatially modulated instabilities of a certain kind.