A note on the uniqueness of the dRGT massive gravity theory

We revisit the problem of gravity coupled to a background metric $\eta_{\mu\nu}$, looking for ghost free interactions. It is known that elimination of the Boulware-Desewr ghost is equivalent to a certain Hessian condition on the interacting potential. We elaborate on this equation and provide a proof that dRGT potential is the unique interaction between the spacetime metric and a background field $\eta_{\mu\nu}$ which is both Lorentz invariant and ghost free. Our approach is fully non-perturbative using only the ADM formulation of general relativity.