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

## TL;DR

This paper proposes a novel approach to ghost-free massive gravity that avoids explicit matrix square roots by using invariants from elementary symmetric polynomials, simplifying the quadratic action construction around Minkowski space.

## Contribution

It introduces a generic method to formulate ghost-free massive gravity using matrix invariants, bypassing the need for explicit square root calculations.

## Key findings

- Constructed quadratic action without matrix square roots.
- Provided a framework for handling non-unique square roots.
- Highlighted potential advantages over standard approaches.

## 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 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 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. Futher details of the latter are deferred to a separate paper.