Stability of massive gravity solutions for holographic conductivity

We consider nonlinear massive gravity with two St\"uckelberg fields in which the diffeomorphism invariance is broken spontaneously in two of the spacetime directions. This theory admits a charged anti--de Sitter black brane solution and has recently been used in the holographic context as a bulk description of a boundary field theory with momentum dissipation. Here we study the stability of the black brane solution. We identify the physical degrees of freedom and determine the healthy regions of the parameter space of the theory. We find that there is a region in parameter space, where the theory suffers from ghost instability. We recognize this as the reason for the instability found in the previous works on holographic conductivity. We also rederive the previous results for the holographic conductivity in a coordinate-independent way.