Bigravity and Lorentz-violating Massive Gravity

TL;DR

This paper explores bigravity theories with Lorentz violation, revealing ghost-free, spherically symmetric solutions with both massless and massive gravitons, and discusses their implications for deviations from General Relativity.

Contribution

It provides a detailed analysis of Lorentz-violating bigravity solutions, including bi-flat and bi-de Sitter cases, and clarifies the linear and non-linear solution discrepancies.

Findings

Existence of Lorentz-breaking bi-flat solutions at large distances.

Linear perturbations include a massless and a massive graviton with two polarizations.

No vDVZ discontinuity; corrections to GR are proportional to graviton mass squared.

Abstract

Bigravity is a natural arena where a non-linear theory of massive gravity can be formulated. If the interaction between the metrics $f$ and $g$ is non-derivative, 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 {\em 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 GR are proportional to the square of the graviton mass, and so there is no vDVZ discontinuity. Surprisingly, the solution of linear theory for a static spherically symmetric source does {\em not} agree with the linearization of any of the known exact solutions. The latter coincide with the standard Schwarzschild-(A)dS 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.