A Note on the Uniqueness of the dRGT Massive Gravity Theory. The D = 3 Case

We revisit the problem of gravity coupled to a background metric ημν, looking for ghost-free interactions. It is known that elimination of the Boulware-Deser ghost is equivalent to a certain Hessian condition on the interacting potential. The resulting equation is a complicated nonlinear partial differential equation. To gain some insight, we consider the d = 3 case and provide evidence that the dRGT potential is the unique interaction between the spacetime metric and a background field ημν which is both Lorentzinvariant and ghost-free. Our approach uses only the ADM formulation of general relativity, without assuming any particular background. One may expect our results to extend to d = 4, but the explicit equation is far more involved.