Arnowitt-Deser-Misner representation and Hamiltonian analysis of covariant renormalizable gravity

TL;DR

This paper analyzes Covariant Renormalizable Gravity (CRG), deriving its ADM form and Hamiltonian structure, revealing it has four propagating degrees of freedom, more than general relativity, due to higher-order derivatives and projectability conditions.

Contribution

It provides the ADM decomposition and Hamiltonian analysis of CRG, identifying its degrees of freedom and comparing them to Hořava-Lifshitz gravity and general relativity.

Findings

CRG contains time derivatives up to fourth order.

The theory has four propagating degrees of freedom.

Two extra modes compared to general relativity, one from higher derivatives and one from projectability.

Abstract

We study the recently proposed Covariant Renormalizable Gravity (CRG), which aims to provide a generally covariant ultraviolet completion of general relativity. We obtain a space-time decomposed form --- an Arnowitt-Deser-Misner (ADM) representation --- of the CRG action. The action is found to contain time derivatives of the gravitational fields up to fourth order. Some ways to reduce the order of these time derivatives are considered. The resulting action is analyzed using the Hamiltonian formalism, which was originally adapted for constrained theories by Dirac. It is shown that the theory has a consistent set of constraints. It is, however, found that the theory exhibits four propagating physical degrees of freedom. This is one degree of freedom more than in Ho\v{r}ava-Lifshitz (HL) gravity and two more propagating modes than in general relativity. One extra physical degree of freedom has its origin in the higher order nature of the CRG action. The other extra propagating mode is a consequence of a projectability condition similarly as in HL gravity. Some additional gauge symmetry may need to be introduced in order to get rid of the extra gravitational degrees of freedom.