  Bimetric variational formalism was recently employed to construct novel
bimetric gravity models. In these models an affine connection is generated by
an additional tensor field which is independent of the physical metric. In this
work we demonstrate how the ADM decomposition can be applied to study such
models and provide some technical intermediate details. Using ADM decomposition
we are able to prove that a linear model is unstable as has previously been
indicated by perturbative analysis. Moreover, we show that it is also very
difficult if not impossible to construct a non-linear model which is ghost-free
within the framework of bimetric variational formalism. However, we demonstrate
that viable models are possible along similar lines of thought. To this end, we
consider a set up in which the affine connection is a variation of the
Levi-Civita one. As a proof of principle we construct a gravity model with a
massless scalar field obtained this way.
