New kinetic terms for massive gravity and multi-gravity: a no-go in vielbein form

We reconsider the possibility of a class of new kinetic terms in the first order (vielbein) formulation of massive gravity and multi-gravity. We find that new degrees of freedom emerge which are not associated with the Boulware-Deser ghost and are intrinsic to the vielbein formulation. These new degrees of freedom are associated with the Lorentz transformations which encode the additional variables contained in the vielbein over the metric. Although they are not guaranteed to be ghostly, they are nevertheless infinitely strongly coupled on Minkowski spacetime and are not part of the spin-2 multiplet. Hence their existence implies the uniqueness of the Einstein-Hilbert term as the kinetic term for a massive graviton.