Linearized gravity in the Randall–Sundrum model with stabilized distance between branes

We consider linearized gravity in the Randall–Sundrum model in which the distance between branes is stabilized by introducing the scalar Goldberger–Wise field. We construct the second variation Lagrangian for fluctuations of gravitational and scalar fields over the background solution and investigate its gauge invariance. We obtain, separate, and solve the corresponding equations of motion. For physical degrees of freedom, we obtain the effective four-dimensional Lagrangian describing the massless graviton, massive gravitons, and the set of massive scalar fields. We also find masses and coupling constants of these fields to the matter on the negative-tension brane.