  We derive the full perturbative equations of motion for the most general
background solutions in ghost-free bimetric theory in its metric formulation.
Clever field redefinitions at the level of fluctuations enable us to circumvent
the problem of varying a square-root matrix appearing in the theory. This
greatly simplifies the expressions for the linear variation of the bimetric
interaction terms. We show that these field redefinitions exist and are
uniquely invertible if and only if the variation of the square-root matrix
itself has a unique solution, which is a requirement for the linearised theory
to be well-defined. As an application of our results we examine the constraint
structure of ghost-free bimetric theory at the level of linear equations of
motion for the first time. We identify a scalar combination of equations which
is responsible for the absence of the Boulware-Deser ghost mode in the theory.
The bimetric scalar constraint is in general not manifestly covariant in its
nature. However, in the massive gravity limit the constraint assumes a
covariant form when one of the interaction parameters is set to zero. For that
case our analysis provides an alternative and almost trivial proof of the
absence of the Boulware-Deser ghost. Our findings generalise previous results
in the metric formulation of massive gravity and also agree with studies of its
vielbein version.
