The scalar constraint which eliminates the ghostly sixth degree of freedom in massive gravity is given explicitly for arbitrary background spacetimes. Many classes of background/parameters can then be identified which correspond to this constraint being trivially satisfied: a result of an extra scalar gauge symmetry. While a vector gauge symmetry (diffeomorphisms) would correspond to a massless graviton, this scalar gauge symmetry is known as a partially massless graviton.

An Einstein spacetime can be thought of as one dominated by a dark energy component, i.e. with a constant Ricci scalar. Previously, various obstructions to propagating a partially massless graviton on non-Einstein spacetimes have been identified - however, these relied on an assumption about the gravitational coupling (that the mass term was at most linear in the background curvature, and the Bach tensor does not vanish), which have been relaxed here. In order to simplify the scalar constraint, the spacetime is assumed to be Ricci symmetric. Finding all of the backgrounds which support a partially massless graviton may provide some insights into extending the theory beyond linear order.