Nonperturbative analysis of the constraints and the positivity of the energy of the complete Hořava theory

We perform a nonperturbative analysis to the Hamiltonian constraint of the lowest-order effective action of the complete Hořava theory, which includes a $({\partial }_i\ln N{)}^2$ term in the Lagrangian. We cast this constraint as a partial differential equation for $N$ and show that the solution exists and is unique under a condition of positivity for the metric and its conjugate momentum. We interpret this condition as the analog of the positivity of the spatial scalar curvature in general relativity. From the analysis we extract several general properties of the solution for $N$: an upper bound on its absolute value and its asymptotic behavior. In particular, we find that the asymptotic behavior is different from that of general relativity, which has consequences on the evolution of the initial data and the calculus of variations. Similarly, we proof the existence and uniqueness of the solution of the equation for the Lagrange multiplier of the theory. We also find a relationship between the expression of the energy and the solution of the Hamiltonian constraint. Using it we prove the positivity of the energy of the effective action under consideration. Minkowski spacetime is obtained from Hořava theory at minimal energy.