Bigravity and Lorentz-violating Massive Gravity

TL;DR

This work analyzes bigravity with non-derivative interactions as a framework for Lorentz-violating massive gravity. It identifies exact Type I bi-flat solutions and proportional-metrics bi-de Sitter backgrounds, and shows that linear perturbations around these backgrounds contain a massless graviton and a two-polarization massive graviton, with no propagating vectors or scalars and no vDVZ discontinuity. Tensor and vector sectors are well-behaved under appropriate mass bounds, while the scalar sector can be free of propagating modes for generic potentials but may exhibit a ghost-condensate-like mode in special cases. The results illuminate how Lorentz violation can stabilize massive gravity theories at linear order, while also exposing subtleties in matching linearized results to non-linear exact solutions and identifying Higuchi-type bounds in de Sitter spaces. The paper thus provides a controlled laboratory to study massive gravity phenomena, perturbative stability, and the interplay between exact and linearized solutions in a multi-metric gravitational theory.

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.