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.

The Hassan–Rosen bimetric theory is an extension of general relativity which considers the interaction between two metric fields defined on the same differentiable manifold. Self-accelerating cosmologies are exact solutions of this theory, and this makes it interesting to explore. We analyze the theory and see if it can provide other physically consistent solutions.

Despite the effort put in studying the theory in recent years, very few exact solutions are known in the literature, and the majority are equivalent to those of general relativity. A valuable approach to try to simplify the field equations and find exact solutions is to impose some spacetime symmetry on the system, e.g., spherical symmetry.

Our study is concerned with symmetries of spacetimes in Hassan–Rosen bimetric theory. Two metrics being present, we investigate what relations exist between their spacetime symmetries. We focus on the isometries of the metrics and clarify when they are the same.

We apply the results in exploring solutions in the Hassan–Rosen theory. We consider maximally symmetric solutions and black hole solutions, and find a previously unknown class of non-stationary spherically symmetric solutions. The existence of the class of non-stationary spherically symmetric solutions disproof a similar statement to Birkhoff's theorem in the Hassan–Rosen bimetric theory. The study of bidiagonal non-rotating black holes sharing the isometries focuses on their properties both at the shared Killing horizon and far from it.

The thesis is a review of these results, and the relevant papers accompany it.

General relativity (GR) is the standard physical theory describing gravitational interactions. All astrophysical and cosmological observations are compatible with its predictions, provided that unknown matter and energy components are included. These are called dark matter and dark energy.

In addition, GR describes the nonlinear self-interaction of a massless spin-2 field. In particle physics, there are both massless and massive fields having spin 0, 1 and 1/2. It is then well-justified to ask whether a mathematically consistent nonlinear theory describing a massive spin-2 field exists.

The Hassan–Rosen bimetric relativity (BR) is a mathematically consistent theory describing the nonlinear interaction between a massless and a massive spin-2 field. These fields are described by two metrics, out of which only one can be directly coupled to us and determines the geometry we probe.

Since it includes GR, BR is an extension of it and provides us with new astrophysical and cosmological solutions. These solutions, which may give hints about the nature of dark matter and dark energy, need to be tested against observations in order to support or falsify the theory. This requires predictions for realistic physical systems. One such system is the spherically symmetric gravitational collapse of a dust cloud, and its study is the overarching motivation behind the thesis.

Studying realistic physical systems in BR requires the solving of the nonlinear equations of motion of the theory. This can be done in two ways: (i) looking for methods that simplify the equations in order to solve them exactly, and (ii) solving the equations numerically.

The studies reviewed in the thesis provide results for both alternatives. In the first case, the results concern spacetime symmetries (e.g., spherical symmetry) and how they affect particular solutions in BR, especially those describing gravitational collapse. In the second case, inspired by the success of numerical relativity, the results initiate the field of numerical bimetric relativity. The simulations provide us with the first hints about how gravitational collapse works in BR.