Non-perturbative analysis of the constraints and the positivity of the energy of the complete Horava theory

TL;DR

This paper performs a non-perturbative analysis of the Hamiltonian constraint in the complete Horava gravity theory, establishing conditions for solution existence, uniqueness, and energy positivity, with implications for the theory's initial data evolution.

Contribution

It provides the first non-perturbative proof of the existence, uniqueness, and positivity of energy in the complete Horava gravity theory, including the analysis of the Lagrange multiplier.

Findings

Existence and uniqueness of the Hamiltonian constraint solution.

Positivity of the energy under certain conditions.

Different asymptotic behavior compared to general relativity.

Abstract

We perform a non-perturbative analysis to the Hamiltonian constraint of the lowest-order effective action of the complete Horava 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 to 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 Horava theory at minimal energy.