Overview
The research addresses the role of non-primary square roots in the mathematical formulation of non-linear dRGT (de Rham, Gabadadze, and Tolley) massive and bimetric gravitational theories. These theories involve matrix square roots, which can suffer from non-uniqueness. Despite existing literature suggesting the exclusive use of principal roots, this study demonstrates that non-primary square roots can maintain general covariance if interpreted correctly. However, a significant issue with non-primary square roots arises in the context of perturbation theory, where their analyticity becomes problematic.
Research Context
Non-linear dRGT massive gravity and bimetric gravity theories rely on the mathematical operation of taking square roots of matrices for their construction. A recognized challenge within this framework is the potential for non-uniqueness in these square roots. Discussions in the field have often concluded that the principal root is preferable, citing concerns that non-primary square roots might violate general covariance. This paper re-examines this specific claim regarding covariance.
Approach
The study analytically investigates the properties of non-primary square roots within the context of dRGT massive and bimetric gravities. It specifically aims to clarify whether these non-primary roots genuinely violate general covariance. Concurrently, the research recalls and re-evaluates an existing observation concerning the behavior of these square roots within perturbation theory. The analysis considers the implications for both matrix-based formulations and those employing elementary symmetric polynomials in the Lagrangian density.
Findings
- Non-primary square roots, when properly understood, are compatible with general covariance in the context of massive and bimetric gravities. This contrasts with some existing claims in the literature that suggest such roots violate general covariance.
- A primary problem associated with non-primary square roots lies within perturbation theory. In a matrix formulation, perturbation theory around these roots effectively does not exist.
- When considering the elementary symmetric polynomials used in the Lagrangian density of these theories, the perturbation theory around non-primary square roots is not analytic.
- Non-principal square roots exhibit a greater propensity to enter the complex domain compared to principal roots.
Why This Matters
The findings clarify a fundamental aspect of the mathematical structure of massive and bimetric gravity theories by demonstrating the covariance of non-primary square roots. This understanding is critical for the consistent theoretical development of these gravitational models. The identified issues with perturbation theory around non-primary roots highlight a significant technical hurdle for further analytical work and potential observational predictions derived from these theories.