Linear spin-2 fields in most general backgrounds

TL;DR

This paper derives linear perturbation equations for ghost-free bimetric theory in general backgrounds, simplifying the analysis via field redefinitions, and confirms the absence of the Boulware-Deser ghost through constraint analysis.

Contribution

It introduces a method to handle square-root matrix variations in bimetric theory and analyzes the constraint structure at the linear level, extending previous results to general backgrounds.

Findings

Derived full linear equations of motion for bimetric backgrounds.

Identified a scalar constraint ensuring ghost freedom.

Confirmed covariant form of the constraint in the massive gravity limit.

Abstract

We derive the full perturbative equations of motion for the most general background solutions in ghost-free bimetric theory in its metric formulation. Clever field redefinitions at the level of fluctuations enable us to circumvent the problem of varying a square-root matrix appearing in the theory. This greatly simplifies the expressions for the linear variation of the bimetric interaction terms. We show that these field redefinitions exist and are uniquely invertible if and only if the variation of the square-root matrix itself has a unique solution, which is a requirement for the linearised theory to be well-defined. As an application of our results we examine the constraint structure of ghost-free bimetric theory at the level of linear equations of motion for the first time. We identify a scalar combination of equations which is responsible for the absence of the Boulware-Deser ghost mode in the theory. The bimetric scalar constraint is in general not manifestly covariant in its nature. However, in the massive gravity limit the constraint assumes a covariant form when one of the interaction parameters is set to zero. For that case our analysis provides an alternative and almost trivial proof of the absence of the Boulware-Deser ghost. Our findings generalise previous results in the metric formulation of massive gravity and also agree with studies of its vielbein version.