Mass spectrum & linear perturbations of ghost-free multi-spin-2 theory

Motivated by the spin-2 nature of gravity, we consider a theory of \( \mathcal{N} \) interacting spin-2 fields, formulated in terms of vielbeins, which has been argued to be free of Boulware-Deser ghost instabilities. This is the only known such theory with non-trivial multi-field interactions. We develop an all-order expansion of the vielbein in terms of the metric and Lorentz field perturbations, enabling an expansion of the action in arbitrary backgrounds to any order. The mass matrix is then analysed around proportional vielbein solutions corresponding to vacuum Einstein spacetimes. It is shown that the theory exhibits a non-tachyonic spectrum with one massless and \( \mathcal{N} \) − 1 massive spin-2 modes, regardless of the sign of the cosmological constant. While the massless mode is universal, the non-zero masses cannot generally be determined analytically, but we obtain the lower and upper bounds on each mass eigenvalue in terms of the parameters of the theory.