Resummation of Massive Gravity

Abstract

We construct four-dimensional covariant non-linear theories of massive gravity which are ghostfree in the decoupling limit to all orders. These theories resum explicitly all the nonlinear terms of an effective field theory of massive gravity. We show that away from the decoupling limit the Hamiltonian constraint is maintained at least up to and including quartic order in non-linearities, hence, excluding the possibility of the Boulware-Deser ghost up to this order. We also show that the same remains true to all orders in a similar toy-model. Introduction:Whether there exist a consistent extension of General Relativity by a mass term is a basic question of a classical field theory. A small graviton mass could also be of a significant physical interest, notably for the cosmological constant problem.A ghost-free linear theory of massive spin-2 -the FierzPauli (FP) model [1] -had been notoriously hard to generalize to the nonlinear level [2]: the Hamiltonian constraint gets lost in general and, as a result, the sixth degree of freedom -the Boulware-Deser (BD) ghostemerges as a mode propagating on otherwise physically meaningful local backgrounds (e.g., on a background of a lump of matter). Part of this problem can be seen in the effective field theory (EFT) approach to massive gravity [3] in the decoupling limit [3,4]. There, the problem manifests itself in the Lagrangian for the helicity-0 component of the massive graviton. This Lagrangian generically contains nonlinear terms with more than two time derivatives. The latter give rise to the sixth degree of freedom on local backgrounds, while in general, these terms lead to the loss of well-posedness of the Cauchy problem for the helicity-0 field theory [3,4].A step forward has been made recently in [5] where it was shown that: (a) the coefficients of the EFT can be chosen so that the decoupling limit Lagrangian is ghostfree; this involves choosing the "appropriate coefficients" order-by-order, and an algorithm was set for this procedure to an arbitrary order; (b) once the "appropriate coefficients" are chosen in the effective Lagrangian, in the decoupling limit only a few terms up to the quartic order survive, all the higher order terms vanish identically. Moreover, the surviving terms are unique as their structure is fixed by symmetries [5,6].In the present work we build on the above two points, and go far beyond them. In particular: (1) We construct Lagrangians that automatically produce the "appropriate coefficients" once expanded in powers of the fields; these give rise to theories that are ghost-free automatically to all orders in the decoupling limit. (2) Using the obtained Lagrangians we study the issue of the BD ghost away from the decoupling limit; we show that the Hamiltonian constraint is maintained at least up to and including

On partially massless bimetric gravity

“…The mass scale m is degenerate with these. The potential is an extension of the 3-parameter massive gravity potential [15] in the formulation of [19]. The equations of motion for g µν and f µν , obtained from (3.1) are given, for example, in [33] on which the present section is based.…”

“…But they are also generically plagued by the Boulware-Deser ghost instability [12,13]. A major breakthrough in this field was the work of [14,15]. Here, the authors developed the nonlinear massive gravity for a flat reference metric f µν = η µν , the dRGT model, and established that it was ghost free in a certain "decoupling limit" analysis [16,17] that proved to be very powerful for the purpose.…”

Nonlinear stability of cosmological solutions in massive gravity

“…The discontinuity can be alleviated by allowing for nonlinear termsà la Vainshtein [4] but, as pointed out in 1972 by Boulware and Deser [5], generic nonlinear massive theories contain an unwanted sixth degree, in addition to the five polarizations of massive spin-2 field, which leads to ghost instability. Since then, it took a long time until a nonlinear extension of the Fierz-Pauli theory without the Boulware-Deser ghost, dubbed the dRGT theory, was found recently [6,7], where the unwanted extra degree is successfully eliminated by construction [8]. However, the absence of the sixth mode does not necessarily imply that the theory is healthy.…”

“…Imposing the absence of the BD ghost in the decoupling limit, the mass part in (1) can be constructed as [6] …”

The Vainshtein mechanism in the cosmic web

“…The non-linear interactions that give an equation of motion containing only up to second derivatives have been constructed imposing a Galilean symmetry [8], and this has led to the discovery of a Boulware-Deser [7] ghost-free massive gravity theory [9]. The Vainshtein mechanism occurs not only in massive gravity [10,11] but also in galileon cosmology [12,13] and braneworld models.…”