Bigravity and Lorentz-violating massive gravity

Bigravity is a natural arena where a nonlinear theory of massive gravity can be formulated. If the interaction between the metrics f and g is nonderivative, spherically symmetric exact solutions can be found. At large distances from the origin, these are generically Lorentz-breaking bi-flat solutions (provided that the corresponding vacuum energies are adjusted appropriately). The spectrum of linearized perturbations around such backgrounds contains a massless as well as a massive graviton, with two physical polarizations each. There are no propagating vectors or scalars, and the theory is ghost free (as happens with certain massive gravities with explicit breaking of Lorentz invariance). At the linearized level, corrections to general relativity are proportional to the square of the graviton mass, and so there is no van Dam-Veltam-Zakharov discontinuity. Surprisingly, the solution of linear theory for a static spherically symmetric source does not agree with the linearization of any of the known exact solutions. The latter coincide with the standard Schwarzschild-(anti)-de Sitter solutions of general relativity, with no corrections at all. Another interesting class of solutions is obtained where f and g are proportional to each other. The case of bi–de Sitter solutions is analyzed in some detail.