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.
Highlights
The theory of General Relativity enjoys a superb agreement with experimental data all over a wide variety of scales
In this paper we present a method of dealing with massive gravity without explicitly taking the square root of the matrix
Tremendous progress has been achieved in the recent years
Summary
The theory of General Relativity enjoys a superb agreement with experimental data all over a wide variety of scales. The early days of massive gravity witnessed an almost detective story which starts from the original paper by Fierz and Pauli [1] which presented the linearised ghost-free massive deformation around flat space, and goes through infamous vDVZ discontinuity [2, 3] of its massless limit, to the potential resolution via Vainshtein mechanism [4, 5], and almost simultaneously to the claim of unavoidable reappearance of the ghost at non-linear level [6], and to the ultimate proposal by de Rham, Gabadadze and Tolley [7, 8, 9, 10, 11]. In this paper we present a method of dealing with massive gravity without explicitly taking the square root of the matrix.
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