# Unusual square roots in the ghost-free theory of massive gravity

## TL;DR

This paper investigates the mathematical intricacies of the square root matrix in ghost-free massive gravity, proposing a new eigenvalue-based formulation to address issues of existence, uniqueness, and perturbation theory.

## Contribution

It introduces an eigenvalue-based approach to massive gravity that overcomes limitations of matrix perturbation theory related to square root ambiguities.

## Key findings

- Discrete and continuous parts of square root freedom identified
- Eigenvalue formulation enables meaningful perturbation analysis
- Addresses non-analytic features in the theory

## Abstract

A crucial building block of the ghost free massive gravity is the square root function of a matrix. This is a problematic entity from the viewpoint of existence and uniqueness properties. 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. It allows for a meaningful discussion of perturbation theory in such cases, even though certain non-analytic features arise.