Consistent massive graviton on arbitrary backgrounds

We obtain the fully covariant linearized field equations for the metric perturbation in the de Rham-Gabadadze-Tolley (dRGT) ghost-free massive gravities. For a subset of these theories, we show that the nondynamical metric that appears in the dRGT setup can be completely eliminated leading to the theory of a massive graviton moving in a single metric. This has a mass term which contains nontrivial contributions of the space-time curvature. We show further how five covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. The five constraints are obtained for a background metric which is arbitrary, i.e. which does not have to obey the background field equations.