Square root of matrices in bimetric theory of gravity

Bimetric gravity is a recently developed theory that provides mass to the graviton via interactional term that includes an additional metric. The important object that arises here is the square root of the product of two metrics. We accurately describe the freedom of choosing a square root of a (non-degenerate) matrix. It has discrete and (in special cases) continuous parts. When continuous freedom is present, the usual perturbation theory in terms of matrices can be critically ill-defined for some choices of the square root. We consider the new formulation of massive and bimetric gravity which deals directly with eigenvalues (in disguise of elementary symmetric polynomials) instead of matrices.