  We consider the issues that arise out of interpreting the ghost-free bimetric
theory as a theory of a spin-2 field coupled to gravity. This requires
identifying a gravitational metric and parameterizing deviations of the
resulting theory from general relativity. To this end, we first consider the
most general bimetric backgrounds for which a massless and a massive spin-2
fluctuation with Fierz-Pauli mass exist. These backgrounds coincide with
solutions in general relativity. Based on this, we obtain nonlinear extensions
of the massive and massless spin-2 fields. The background value of the
nonlinear massive field parameterizes generic deviations of the bimetric theory
from GR. It is also shown that the nonlinear massless field does not have
standard ghost-free matter couplings, and hence cannot represent the
gravitational metric. However, an appropriate gravitational metric can still be
identified in the weak gravity limit. Hence in the presence of other neutral
spin-2 fields, the weak gravity limit is crucial for compatibility with general
relativity. We also write down the action in terms of the nonlinear massive
spin-2 field and obtain its ghost-free couplings to matter. The discussion is
then generalized to multimetric theories.
