Non-primary square roots in massive gravity

Abstract

Non-linear dRGT massive and bimetric gravities are complicated theories constructed in terms of square roots of matrices. Apart from the technical issues of successfully working with such square roots, there is also a problem of their non-uniqueness. There are claims in the literature that one should better use the principal root. This is a very reasonable conclusion. However, the motivation they give for it is that otherwise there would be non-primary square roots violating the general covariance. In this paper, I would like to show that, if properly understood, the non-primary square roots are also perfectly covariant. At the same time, I recall the relatively old observation that the real problem with such square roots lies in perturbation theory around them. In terms of matrices, it simply does not exist. In terms of the elementary symmetric polynomials used in the Lagrangian density, it is not analytic. Moreover, the non-principal square roots are more prone to getting into the complex domain.