A note on the uniqueness of the dRGT massive gravity theory

TL;DR

This paper proves that the dRGT potential is the only Lorentz-invariant, ghost-free interaction between a spacetime metric and a background metric, using a non-perturbative ADM approach.

Contribution

It provides a non-perturbative proof of the uniqueness of the dRGT potential for ghost-free massive gravity with background metrics.

Findings

dRGT potential is unique under Lorentz invariance and ghost-free conditions

The proof is non-perturbative and uses ADM formulation

Elimination of Boulware-Deser ghost is linked to a Hessian condition

Abstract

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.