Vielbein to the rescue? Breaking the symmetric vielbein condition in massive gravity and multigravity

Abstract

Nonminimal matter couplings have recently been considered in the context of massive gravity and multigravity. These couplings are free of the Boulware-Deser ghost in the decoupling limit and can thus be considered within an effective field theory setup. Beyond the decoupling limit the ghost was shown to reemerge in the metric formulation of the theory. Recently it was argued that this pathology is absent when formulated in terms of unconstrained vielbeins. We investigate this possibility and show that the BoulwareDeser ghost is always present beyond the decoupling limit in any dimension larger than 2. We also show that the metric and vielbein formulations have an identical ghost-free decoupling limit. Finally we extend these arguments to more generic multigravity theories and argue that for any dimension larger than 2 a ghost is also present in the vielbein formulation whenever the symmetric vielbein condition is spoiled and the equivalence with the metric formulation is lost.

Cosmology with three interacting spin-2 fields

“…(27) and (30) suggest, the dynamics of trimetric cosmology are captured by the scale-factor ratios r 1 and r 2 . Thus, we need to find an expression for r i in order to be able to analyze the background cosmology of star trigravity.…”

“…This provides an immediate avenue for trimetric gravity: we could couple each pair of metrics via the ghost-free potential, leading to a cycle of interactions. However, such cycles turn out to be plagued by the Boulware-Deser ghost [25][26][27]. Therefore we are forced to consider breaking the cycle into a line, i.e., there must be one pair of metrics which do not directly interact with each other.…”

Structures in multiple spin-2 interactions

“…For instance, the results of Ref. [32,36] obtained for the vierbein formulation of bimetric theory imply the absence of additional couplings for several metrics. 4 Analogous arguments given purely in terms of metrics can be found in Ref.…”

“…The centre coupling is visualised in Figure 2. It is not possible to introduce additional couplings between two of the "satellite metrics" f i µν because such terms would destroy the constraints that remove the Boulware-Deser ghosts [32,34,36]. In general, any type of "loop coupling" is forbidden in multimetric theory.…”

“…[33][34][35]). It later turned out that in general such couplings suffer from the BoulwareDeser ghost [36], while some particular interaction terms considered in Ref. [32] still avoid the instability.…”