Confirmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity (1111.2070v1)

Abstract: In massive gravity and in bimetric theories of gravity, two constraints are needed to eliminate the two phase-space degrees of freedom of the Boulware-Deser ghost. For recently proposed non-linear theories, a Hamiltonian constraint has been shown to exist and an associated secondary constraint was argued to arise as well. In this paper we explicitly demonstrate the existence of the secondary constraint. Thus the Boulware-Deser ghost is completely absent from these non-linear massive gravity theories and from the corresponding bimetric theories.

• The paper rigorously confirms that introducing a secondary constraint eliminates the Boulware-Deser ghost in massive gravity theories.
• It employs an ADM decomposition and introduces new shift-like variables to simplify the Hamiltonian analysis.
• The validation extends to bimetric gravity, ensuring consistency and stability in modified gravitational frameworks.

Insightful Overview of Constraints in Massive Gravity and Bimetric Gravity

The paper by S. F. Hassan and Rachel A. Rosen presents a detailed analysis of constraint dynamics in massive gravity and bimetric theories, focusing specifically on the elimination of the Boulware-Deser ghost mode. This theoretical construct, central to many developments in modified gravity, historically posed challenges due to potential instabilities introduced by ghost modes. The authors address the validation of associated secondary constraints required for the consistent formulation of these theories.

Key Contributions

1. Hamiltonian Constraint Validation: The paper comprehensively revisits the Hamiltonian constraint within the context of massive gravity. It builds upon prior results to provide a rigorous derivation of the secondary constraint, thus confirming the absence of the Boulware-Deser ghost. The analysis asserts the vanishing of certain Poisson brackets, essential for maintaining a consistent phase space devoid of ghost dynamics.
2. Introduction of New Variables: Utilizing an ADM (Arnowitt-Deser-Misner) decomposition, the authors introduce novel shift-like variables to simplify the problem. This reformulation aids in manifesting the Hamiltonian constraint and facilitates a clearer demonstration of its implications.
3. Verification in Bimetric Gravity: The extension of these results to bimetric gravity formulations marks a significant validation step, as the promotion of the reference metric to a dynamic field adds complexity. The paper successfully generalizes the absence of the ghost field, ensuring that these extended theories maintain stability and consistency with theoretical predictions.

Theoretical Implications and Future Prospects

The explicit proof of secondary constraints removes significant obstacles for massive gravity theories, providing a clearer path to further theoretical and observational studies of modified gravity. The elimination of the Boulware-Deser ghost without introducing additional inconsistencies ensures that these frameworks remain viable contenders within the broader spectrum of gravitational theories, particularly when addressing cosmological observations such as dark energy and cosmic acceleration.

Conclusions

This paper resolves critical questions regarding the stability of non-linear massive gravity and its bimetric counterpart, confirming theoretically that they can be made consistent without ghost degrees of freedom. As such, these findings position massive and bimetric gravity frameworks as robust alternatives to General Relativity, ripe for experimental exploration. Future research may focus on the exploration of additional solutions within these theories and their potential observable effects in astrophysical and cosmological contexts. This includes studying perturbative modes around different backgrounds and investigating their phenomenological consequences.

The methodological approaches and rigorous analyses detailed in this paper will serve as foundational references within ongoing research in gravitational theories and may catalyze further advancements within the field.