Ghost-free Massive Gravity with a General Reference Metric (1109.3230v2)

Abstract: Theories of massive gravity inevitably include an auxiliary reference metric. Generically, they also contain an inconsistency known as the Boulware-Deser ghost. Recently, a family of non-linear massive gravity actions, formulated with a flat reference metric, were proposed and shown to be ghost free at the complete non-linear level. In this paper we consider these non-linear massive gravity actions but now formulated with a general reference metric. We extend the proof of the absence of the Boulware-Deser ghost to this case. The analysis is carried out in the ADM formalism at the complete non-linear level. We show that in these models there always exists a Hamiltonian constraint which, with an associated secondary constraint, eliminates the ghost. This result considerably extends the range of known consistent non-linear massive gravity theories. In addition, these theories can also be used to describe a massive spin-2 field in an arbitrary, fixed gravitational background. We also discuss the positivity of the Hamiltonian.

• The paper introduces a non-linear Hamiltonian framework that eliminates the Boulware-Deser ghost using a generalized reference metric.
• It employs the ADM formalism and symmetric polynomial functions to ensure only the legitimate five graviton polarizations are present.
• The study extends massive gravity theories to arbitrary reference metrics, paving the way for bi-metric models and new cosmological applications.

Ghost-Free Massive Gravity with a General Reference Metric

The paper under consideration addresses significant challenges in the field of massive gravity theories, particularly focusing on the avoidance of the Boulware-Deser (BD) ghost. At the core of these theories is the need for an additional rank-2 symmetric tensor, referred to as the reference metric, which typically has been considered in its simplest form as flat. This research extends the framework to encompass a general reference metric and proves that these theories maintain consistency and freedom from ghost instabilities, even beyond specific cases like flat metrics.

The Problem of Boulware-Deser Ghost

A long-standing issue in theories of massive gravity is the Boulware-Deser ghost anomaly, which arises at non-linear levels, potentially introducing a sixth, ghost mode with negative kinetic energy. Boulware and Deser's foundational work highlighted the inherent difficulties in removing this ghost mode beyond the constraints successfully eliminated in the linear Fierz-Pauli theory. This work instead posits and proves conditions under which a non-linear Hamiltonian constraint exists, thereby inherently removing the ghost without the need to linearize in terms of the scalar lapse and shift functions across all distributions of the metric.

Key Findings

1. Non-Linear Formulation: Employing a generalized reference metric within a non-linear framework, the paper demonstrates the absence of the BD ghost via Hamiltonian and associated secondary constraints. These constraints ensure that only the legitimate five polarizations of a massive graviton are involved, maintaining physical consistency.
2. Methodology: Utilizing the Arnowitt-Deser-Misner (ADM) formalism, the authors showcase how appropriate conditions on the potential terms can consistently eliminate ghost contributions at the non-linear level. With the formulation expressed through a set of symmetric polynomial functions of the eigenvalues of the metric's square-root, the theory accommodates a broad class of reference metrics.
3. Reformulation and Generalization: Building on prior work which addressed flat metrics, this paper broadens applicability to arbitrary non-dynamical metrics, establishing a theoretically satisfying ground that's not constrained to de Sitter or Minkowski frameworks. This extension aids in addressing critiques and potential solution classes previously thought to be invalid within non-linear massive gravity landscapes.
4. Analytical Results and Implications: The derived results carry implications for both cosmologies in novel gravitational backgrounds and potential applications in other massive spin-2 field theories. Empirical stability analysis shows that, while some setups still carry instabilities, these are separable from ghost-like behaviour and pertain to specific parameter regions rather than general breaches of the theory's foundational consistency.

Future Directions

By confirming the ghost-free consistency of these more generalized theories, the paper opens avenues for the development of bi-metric gravity models where the reference metric is allowed dynamics, resembling proposals for bimetric gravity systems. It also invites comparisons and integration efforts with AdS/CFT duality and string theory frameworks, where emergence within an AdS structure has been proposed. Future work may address the promotion of fμν to a dynamic metric, diving deeper into the realization of such systems from fundamental string theory perspectives.

In conclusion, this work represents a thorough advancement in proving the viability of ghost-free massive gravity theories, particularly through the strategic formulation of the Hamiltonian and its constraints across generalized backgrounds. This approach not only resolves theoretical bottlenecks but also enriches the toolkit for cosmologists exploring gravitational theories with massive spin-2 fields.