Diffeomorphism invariance and diffeomorphism generation in the Horava-Lifshitz gravity

TL;DR

This paper investigates the diffeomorphism invariance of Horava-Lifshitz gravity, showing it is preserved under certain conditions and that the theory's constraints generate all diffeomorphisms of General Relativity.

Contribution

It clarifies the conditions under which Horava-Lifshitz gravity maintains diffeomorphism invariance and demonstrates that its constraints generate all GR diffeomorphisms on-shell.

Findings

Diffeomorphism invariance is preserved when Levi-Civita terms are neglected.

Constraints in Horava-Lifshitz gravity generate all GR diffeomorphisms on-shell.

The theory's invariance depends on specific terms in the action.

Abstract

This paper is intended to study diffeomorphism invariance and diffeomorphism generation in the modified theory of gravity proposed by Horava. Firstly, we demonstrate that the theory does not lose diffeomorphism invariance due to the parameter $\lambda$, as it was previously believed. However, we show that the presence of terms containing the Levi-Civita symbol in the original proposal of Horava makes the theory diffeomorphism dependent. By neglecting such terms, what returns fully diffeomorphism invariance to the action, we obtain the equations of motion. Secondly, in the Hamiltonian formalism, we calculate the transformations generated by some of the constraints of the theory. Then, we prove that all diffeomorphisms of General Relativity are generated, on the energy shell, by the constraints of the Horava-Lifshitz gravity.