The ghost-free bimetric theory describes interactions of gravity with another spin-2 field in terms of two Lorentzian metrics. However, if the two metrics do not admit compatible notions of space and time, the formulation of the initial value problem becomes problematic. Furthermore, the interaction potential is given in terms of the square root of a matrix which is in general nonunique and possibly nonreal. In this paper we show that both these issues are evaded by requiring reality and general covariance of the equations. First we prove that the reality of the square root matrix leads to a classification of the allowed metrics in terms of the intersections of their null cones. Then, the requirement of general covariance further restricts the allowed metrics to geometries that admit compatible notions of space and time. It also selects a unique definition of the square root matrix. The restrictions are compatible with the equations of motion. These results ensure that the ghost-free bimetric theory can be defined unambiguously and that the two metrics always admit compatible 3+1 decompositions, at least locally. In particular, these considerations rule out certain solutions of massive gravity with locally Closed Causal Curves, which have been used to argue that the theory is acausal.

We consider the Hassan-Rosen bimetric field equations in vacuum when the two metrics share a single common null direction in a spherically symmetric configuration. By solving these equations, we obtain a class of exact solutions of the generalized Vaidya type parametrized by an arbitrary function. Besides not being asymptotically flat, the found solutions are nonstationary admitting only three global spacelike Killing vector fields which are the generators of spatial rotations. Hence, these are spherically symmetric bimetric vacuum solutions with the minimal number of isometries. The absence of staticity formally disproves an analogue statement to Birkhoff's theorem in the ghost-free bimetric theory which would state that a spherically symmetric solution is necessarily static in empty space.

We study general properties of static and spherically symmetric bidiagonal black holes in Hassan-Rosen bimetric theory by means of a new method. In particular, we explore the behavior of the black hole solutions both at the common Killing horizon and at the large radii. The former study was never done before and leads to a new classification for black holes within the bidiagonal ansatz. The latter study shows that, among the great variety of the black hole solutions, the only solutions converging to Minkowski, anti-de Sitter, and de Sitter spacetimes at large radii are those of general relativity, i.e., the Schwarzschild, Schwarzschild-anti-de Sitter and Schwarzschild-de Sitter solutions. Moreover, we present a proposition, whose validity is not limited to black hole solutions, which establishes the relation between the curvature singularities of the two metrics and the invertibility of their interaction potential.