The Boulware-Deser mode in 3D first-order massive gravity

Abstract

Massive gravity in three dimensions accepts several different formulations. Recently, the 3-dimensional bigravity dRGT model in first order form, Zwei-Dreibein gravity, was considered by Bergshoeff et al. and it was argued that the Boulware-Deser mode is killed by extra constraints. We revisit this assertion and conclude that there are sectors on the space of initial conditions, or subsets of the most general such model, where this mode is absent. But, generically, the theory does carry 3 degrees of freedom and thus the Boulware-Deser mode is still active. Our results also sheds light on the equivalence between metric and vierbein formulations of dRGT model.The search for a well-defined, unitary, stable, massive version of general relativity has seen huge interest in recent years (for a review see [17] A particularly simple and nice formulation of dRGT gravity was put forward in [18] (see also [11,12] for a discussion on the equivalence between metric and vielbein formulations). The action is built using vielbeins 1-forms and their corresponding 2-forms curvatures. A three dimensional version of this formulation, which can shed light on the four dimensional one, has recently been considered in [4]. The action iswhereê a andl a are two independent dreibeins. Here and henceforth wedge product are implicit. The connections are denoted byŵ a andπ a with curvatureŝAll hatted quantities are spacetime forms. The corresponding spatial forms will be denoted by the same letter without the hat. Latin indexes are raised and lowered with Minkowski metric η ab and η ab . For simplicity we do not incorporate cosmological constants at each sector. k 1 and k 2 are free parameters. It was argued in [4] that (1) does not carry a BoulwareDeser mode, in agreement with the 4-dimensional claims (mostly based on the metric formulation, see however [1,12,18]). The goal of this Letter is to critically analyze this issue. Our conclusion will be that the BoulwareDeser mode is generically still active in the formulation (1) even though there are indeed subcases where it is absent.The simplicity of working in three dimensions is seen by the fact that the action (1) is already in Hamiltonian form. One only needs to perform a 2+1 decomposition of forms,êand likewise forŵ a µ dx µ ,π a µ dx µ . The action in the 2+1 decomposition becomeswhere one can read the symplectic structure in a straightforward way. Here 'dot' stands for time derivative andNote that we still use form notation on the 2-dimensional spatial manifold. The spatial fields {e a i , w b j } and {ℓ a i , π b j } form 12 canonical pairs, while the temporal components e a 0 , ℓ a 0 , w a 0 , π a 0 are 12 Lagrange multipliers. This property is characteristic of generally covariant systems and, as we shall remark below, has important consequences on the consistency algorithm. Let us do a first counting of degrees of freedom based on the number of canonical variables and constraints (we shall argue below that there are no secondary constraints in the most generic case). There are 24 ca...

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