[Paper Review] Massive graviton on arbitrary background: derivation, syzygies, applications

Laura Bernard, Cédric Deffayet+1 more|arXiv (Cornell University)|Apr 16, 2015

Cosmology and Gravitation Theories47 references3 citations

TL;DR

This paper provides a fully covariant derivation of the quadratic action and linearized equations of motion for a massive graviton on an arbitrary background metric, eliminating the need for a reference metric by expressing the theory in terms of a single dynamical metric. The key contribution is a covariant proof that the graviton propagates at most five degrees of freedom on any background, with five constraints—among which one ensures tracelessness and removes the Boulware-Deser ghost—derived from curvature-dependent identities (syzygies) involving the background curvature and graviton fields.

ABSTRACT

We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a "reference metric" which is present in the non perturbative formulation. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lema\^ıtre-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reach our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.

Motivation & Objective

• To derive a fully covariant formulation of the massive graviton action and equations of motion on an arbitrary background metric, without relying on a separate reference metric.
• To prove that the massive graviton propagates at most five physical degrees of freedom on any background, including curved spacetimes.
• To identify and derive five covariant constraints—among them a curvature-dependent one that eliminates the Boulware-Deser ghost—valid even when the background does not satisfy Einstein's equations.
• To correct and clarify previous results on Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes in massive gravity.
• To uncover and establish nontrivial syzygies (identities) between the graviton field, its derivatives, and the background curvature, which are essential for the consistency of the theory.

Proposed method

• Starts from the de Rham-Gabadadze-Tolley (dRGT) family of ghost-free massive gravity theories and focuses on a simplified model to eliminate the reference metric.
• Derives the quadratic action and linearized field equations in a fully covariant form using the background metric as the sole geometric structure.
• Applies a Hamiltonian-like analysis to identify five covariant constraints, including one that enforces tracelessness of the graviton and removes the Boulware-Deser ghost.
• Derives nontrivial syzygies involving the graviton field, its derivatives, and the Riemann curvature of the background metric, which are crucial for consistency.
• Validates the results by showing that the constraints become trivial in Einstein space-times and corrects prior results in FLRW backgrounds.
• Uses alternative derivations via background field equations and formal inversion of the linearized operator to cross-check the field equations and mass matrix.

Experimental results

Research questions

• RQ1 Can the massive graviton action and equations of motion be formulated in a fully covariant way on an arbitrary background metric without a reference metric?
• RQ2 What are the constraints that ensure the massive graviton has at most five physical degrees of freedom on a generic curved background?
• RQ3 How does the Boulware-Deser ghost get removed in a fully covariant, background-dependent formulation of massive gravity?
• RQ4 What role does the background curvature play in the mass term and in the structure of the field equations?
• RQ5 Are there backgrounds for which the dRGT equations cannot be unambiguously linearized, and if so, what are the conditions?

Key findings

• The massive graviton action and linearized field equations are derived in a fully covariant form using only the physical background metric, eliminating the need for a separate reference metric.
• Five covariant constraints are derived that ensure the graviton has at most five physical degrees of freedom, with one constraint explicitly removing the Boulware-Deser ghost via curvature-dependent terms.
• The theory is consistent on arbitrary backgrounds, even when the background does not satisfy the Einstein field equations, and the constraints remain nontrivial in general.
• In Einstein space-times, the five constraints become trivial, confirming consistency with known results.
• For Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, the paper corrects earlier results in the literature, providing a consistent linearized theory.
• Nontrivial syzygies between the graviton field, its derivatives, and the background curvature are derived, which are essential for the consistency of the equations and have independent mathematical interest.