Consistency of the Hamiltonian formulation of the lowest-order effective action of the complete Hořava theory

Abstract

We perform the Hamiltonian analysis for the lowest-order effective action, up to second order in derivatives, of the complete Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory. The model includes the invariant terms that depend on ${\ensuremath{\partial}}_{i}\mathrm{ln}N$ proposed by Blas, Pujol\`as, and Sibiryakov. We show that the algebra of constraints closes. The Hamiltonian constraint is of second-class behavior and it can be regarded as an elliptic partial differential equation for $N$. The linearized version of this equation is a Poisson equation for $N$ that can be solved consistently. The preservation in time of the Hamiltonian constraint yields an equation that can be consistently solved for a Lagrange multiplier of the theory. The model has six propagating degrees of freedom in the phase space, corresponding to three even physical modes. When compared with the $\ensuremath{\lambda}R$ model studied by us in a previous paper, it lacks two second-class constraints, which leads to the extra even mode.