Dealing with ghost-free massive gravity without explicit square roots of matrices

Abstract

In this paper we entertain a simple idea that the action of ghost free massive gravity (in metric formulation) depends not on the full structure of the square root of a matrix but rather on its invariants given by the elementary symmetric polynomials of the eigenvalues. In particular, we show how one can construct the quadratic action around Minkowski spacetime without ever taking the square root of the perturbed matrix. The method is however absolutely generic. And it also contains the full information on possible non-standard square roots coming from intrinsic non-uniqueness of the procedure. In passing, we mention some hard problems of those apocryphal square roots in the standard approach which might be better tackled with our method. The details of the latter are however deferred to a separate paper.