We analyze a bi-gravity model based on the first order formalism, having as
fundamental variables two tetrads but only one Lorentz connection. We show that
on a large class of backgrounds its linearization agrees with general
relativity. At the non-linear level, additional degrees of freedom appear, and
we reveal the mechanism hiding them around the special backgrounds. We further
argue that they do not contain a massive graviton, nor the Boulware-Deser
ghost. The model thus propagates only one graviton, whereas the nature of the
additional degrees of freedom remains to be investigated. We also present a
foliation-preserving deformation of the model, which keeps all symmetries
except time diffeomorphisms and has three degrees of freedom.Comment: 29 page