The ghost-free bimetric theory is an extension to general relativity where two metric tensors are
used instead of one. A priori, the two metrics may not have compatible notions of space and time,
which makes the formulation of the initial-value problem problematic. Moreover, the metrics are
coupled through a specific ghost-free interaction term that is nonunique and possibly nonreal. We
prove that the reality of the bimetric potential leads to a classification of the allowed configurations
of the two metrics in terms of the intersections of their null cones. Then, the equations of motion
and general covariance are enough to restrict down the allowed configurations to metrics that admit
compatible notions of space and time, and furthermore, lead to a unique definition of the potential.
This ensures that the ghost-free bimetric theory can be defined unambiguously. In addition, we
apply the results on spherically symmetric spacetimes. First, we explore the behavior of the black
hole solutions both at the common Killing horizon and at the large radii. The study leads to a
new classification for black holes within the bidiagonal ansatz. Finally, we consider the bimetric
field equations in vacuum when the two metrics share a single common null direction. We obtain
a class of exact solutions of the generalized Vaidya type parametrized by an arbitrary function.
The found solutions are nonstationary and thus nonstatic, which formally disproves an analogous
statement to Birkhoff’s theorem in the ghost-free bimetric theory.