On Consistent Theories of Massive Spin-2 Fields Coupled to Gravity

TL;DR

This work analyzes the ghost-free ghost-free bimetric theory as a theory of a massive spin-2 field interacting with gravity, focusing on identifying the gravitational metric and characterizing deviations from GR. It constructs nonlinear extensions of the linear massless and massive spin-2 modes and shows that the nonlinear massless field cannot couple ghost-free to matter, while the gravitational metric at low energies is best associated with the $g$-sector in the weak-gravity limit. The authors then formulate the theory in terms of the gravitational metric $g$ and the nonlinear massive field $M$, derive the ghost-free couplings to matter, and extend the construction to multimetric theories with multiple massive spin-2 fields. The results clarify how bimetric theories can mimic GR in appropriate limits while capturing new spin-2 dynamics, and provide a concrete framework for exploring deviations from GR in both single- and multi-spin-2 settings.

Abstract

We consider the issues that arise out of interpreting the ghost-free bimetric theory as a theory of a spin-2 field coupled to gravity. This requires identifying a gravitational metric and parameterizing deviations of the resulting theory from general relativity. To this end, we first consider the most general bimetric backgrounds for which a massless and a massive spin-2 fluctuation with Fierz-Pauli mass exist. These backgrounds coincide with solutions in general relativity. Based on this, we obtain nonlinear extensions of the massive and massless spin-2 fields. The background value of the nonlinear massive field parameterizes generic deviations of the bimetric theory from GR. It is also shown that the nonlinear massless field does not have standard ghost-free matter couplings, and hence cannot represent the gravitational metric. However, an appropriate gravitational metric can still be identified in the weak gravity limit. Hence in the presence of other neutral spin-2 fields, the weak gravity limit is crucial for compatibility with general relativity. We also write down the action in terms of the nonlinear massive spin-2 field and obtain its ghost-free couplings to matter. The discussion is then generalized to multimetric theories.