Reduced massive gravity with two Stückelberg fields

We consider the nonlinear massive gravity as a theory of a number of Stückelberg scalar fields minimally coupled to the Einstein-Hilbert gravity and argue that the counting of degrees of freedom can be done for scalar theory and gravity separately. In this paper we investigate the system with only two Stückelberg scalar fields. In this case we find the analytic expression for the determinant of the kinetic matrix of the scalar field Lagrangian and perform the full constraint analysis. In $1+1$ space-time dimensions, the theory corresponds to the full nonlinear massive gravity, and this determinant vanishes identically. In this case we find two first-class constraints and present the corresponding gauge symmetry of the theory which eliminates both scalar degrees of freedom. In $3+1$ dimensions, the determinant of the kinetic matrix does not vanish identically and, for generic initial conditions, both scalar fields are propagating.

Figure 1

Some members of the family of regular solutions [Eq. (3.23)] (solid), which approach a given singular solution ${\varphi }^0=0$ (dashed) in the limit $t\to \pm \infty$.