Recent developments in bimetric theory

This review is dedicated to recent progress in the field of classical, interacting, massive spin-2 theories, with a focus on ghost-free bimetric theory. We will outline its history and its development as a nontrivial extension and generalisation of nonlinear massive gravity. We present a detailed discussion of the consistency proofs of both theories, before we review Einstein solutions to the bimetric equations of motion in vacuum as well as the resulting mass spectrum. We introduce couplings to matter and then discuss the general relativity and massive gravity limits of bimetric theory, which correspond to decoupling the massive or the massless spin-2 field from the matter sector, respectively. More general classical solutions are reviewed and the present status of bimetric cosmology is summarised. An interesting corner in the bimetric parameter space which could potentially give rise to a nonlinear theory for partially massless spin-2 fields is also discussed. Relations to higher-curvature theories of gravity are explained and finally we give an overview of possible extensions of the theory and review its formulation in terms of vielbeins.Comment: 124 pages, 4 figures; minor changes, corrected typos; matches published versio

On extended symmetries for the Galileon

We investigate a large class of infinitesimal, but fully nonlinear in the field, transformations of the Galileon and search for extended symmetries. The transformations involve powers of the coordinates $x$ and the field $\pi$ up to any finite order $N$. Up to quadratic order the structure of these symmetry transformations is the unique generalisation of both the infinitesimal version of the standard Galileon shift symmetry as well as a recently discovered infinitesimal extension of this symmetry. The only higher-order extensions of this symmetry we recover are (`Galileon dual' versions of) symmetries of the standard kinetic term.Comment: Latex, 5 pages, conclusions change

Partially Massless Graviton on Beyond Einstein Spacetimes

We show that a partially massless graviton can propagate on a large set of spacetimes which are not Einstein spacetimes. Starting from a recently constructed theory for a massive graviton that propagates the correct number of degrees of freedom on an arbitrary spacetime, we first give the full explicit form of the scalar constraint responsible for the absence of a sixth degree of freedom. We then spell out generic conditions for the constraint to be identically satisfied, so that there is a scalar gauge symmetry which makes the graviton partially massless. These simplify if one assumes that spacetime is Ricci symmetric. Under this assumption, we find explicit non-Einstein spacetimes allowing for the propagation of a partially massless graviton.Comment: 17 pages. v2 error in eq. 31 and its propagation correcte

Particular Solutions in Bimetric Theory and Their Implications

Ghost-free bimetric theory can describe gravity in the presence of an extra spin-2 field. We study certain aspects of dynamics in this theory: (1) It is shown that if either of the metrics is an Einstein solution then the other is always forced to be Einstein, too. For a class of bimetric models this constraint is stronger and as soon as one metric is Einstein, the other metric is forced to be proportional to it. As a consequence, the models in this class avoid a branch of pathological solutions that exhibit determinant singularities or nonlinear ghosts. These constraints persists in a generalized form when sources are included, but are destroyed in the massive gravity limit of the theory. (2) For another class of bimetric models, we show the existence of solutions that do not admit a massive gravity limit. A bimetric model that could exhibit a nonlinear version of "partially massless" symmetry belongs to both these classes. It is argued that if such a model exits, its symmetry will not survive in the massive gravity limit.Comment: Latex, 18 pages. Published versio

Extended Weyl Invariance in a Bimetric Model and Partial Masslessness

We revisit a particular ghost-free bimetric model which is related to both partial masslessness (PM) and conformal gravity. Linearly, the model propagates six instead of seven degrees of freedom not only around de Sitter but also around flat spacetime. Nonlinearly, the equations of motion can be recast in the form of expansions in powers of curvatures, and exhibit a remarkable amount of structure. In this form, the equations are shown to be invariant under scalar gauge transformations, at least up to six orders in derivatives, the lowest order term being a Weyl scaling of the metrics. The terms at two-derivative order reproduce the usual PM gauge transformations on de Sitter backgrounds. At the four-derivative order, a potential obstruction that could destroy the symmetry is shown to vanish. This in turn guarantees the gauge invariance to at least six-orders in derivatives. This is equivalent to adding up to 10-derivative corrections to conformal gravity. More generally, we outline a procedure for constructing the gauge transformations order by order as an expansion in derivatives and comment on the validity and limitations of the procedure. We also discuss recent arguments against the existence of a PM gauge symmetry in bimetric theory and show that, at least in their present form, they are evaded by the model considered here. Finally, we argue that a bimetric approach to PM theory is more promising than one based on the existence of a fundamental PM field.Comment: Latex, 35 pages. Matches published versio

Linear spin-2 fields in most general backgrounds

We derive the full perturbative equations of motion for the most general background solutions in ghost-free bimetric theory in its metric formulation. Clever field redefinitions at the level of fluctuations enable us to circumvent the problem of varying a square-root matrix appearing in the theory. This greatly simplifies the expressions for the linear variation of the bimetric interaction terms. We show that these field redefinitions exist and are uniquely invertible if and only if the variation of the square-root matrix itself has a unique solution, which is a requirement for the linearised theory to be well-defined. As an application of our results we examine the constraint structure of ghost-free bimetric theory at the level of linear equations of motion for the first time. We identify a scalar combination of equations which is responsible for the absence of the Boulware-Deser ghost mode in the theory. The bimetric scalar constraint is in general not manifestly covariant in its nature. However, in the massive gravity limit the constraint assumes a covariant form when one of the interaction parameters is set to zero. For that case our analysis provides an alternative and almost trivial proof of the absence of the Boulware-Deser ghost. Our findings generalise previous results in the metric formulation of massive gravity and also agree with studies of its vielbein version.Comment: Latex, 20 pages. Matches published versio

Consistent massive graviton on arbitrary backgrounds

We obtain the fully covariant linearized field equations for the metric perturbation in the de Rham-Gabadadze-Tolley (dRGT) ghost free massive gravities. For a subset of these theories, we show that the non dynamical metric that appears in the dRGT setup can be completely eliminated leading to the theory of a massive graviton moving in a single metric. This has a mass term which contains non trivial contributions of the space-time curvature. 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. The 5 constraints are obtained for a background metric which is arbitrary, i.e. which does not have to obey the background field equations.Comment: Latex, 5 pages. Matches published versio

Higher Derivative Gravity and Conformal Gravity From Bimetric and Partially Massless Bimetric Theory

In this paper we establish the correspondence between ghost-free bimetric theory and a class of higher derivative gravity actions, including conformal gravity and New Massive Gravity. We also characterize the relation between the respective equations of motion and classical solutions. We illustrate that, in this framework, the spin-2 ghost of higher derivative gravity is an artifact of the truncation to a 4-derivative theory. The analysis also gives a relation between the proposed partially massless (PM) bimetric theory and conformal gravity, showing, in particular, the equivalence of their equations of motion at the 4-derivative level. For the PM bimetric theory this provides further evidence for the existence of an extra gauge symmetry and the associated loss of a propagating mode away from de Sitter backgrounds. The new symmetry is an extension of Weyl symmetry which also suggests the PM bimetric theory as a ghost-free completion of conformal gravity.Comment: Latex, 34 pages; minor comments and note added, matches published versio