Covariant St\"uckelberg analysis of de Rham-Gabadadze-Tolley massive gravity with a general fiducial metric

The St\"uckelberg analysis of nonlinear massive gravity in the presence of a general fiducial metric is investigated. We develop a "covariant" formalism for the St\"uckelberg expansion by working with a local inertial frame, through which helicity modes can be characterized correctly. Within this covariant approach, an extended $\Lambda_3$ decoupling limit analysis can be consistently performed, which keeps $\bar{R}_{\mu\nu\rho\sigma}/m^2$ fixed with $\bar{R}_{\mu\nu\rho\sigma}$ the Riemann tensor of the fiducial metric. In this extended decoupling limit, the scalar mode $\pi$ acquires self-interactions due to the presence of the curvature of the fiducial metric. However, the equation of motion for $\pi$ remains of second order in derivatives, which extends the understanding of the absence of the Boulware Deser ghost in the case of a flat fiducial metric.