Hamiltonian analysis of spatially covariant gravity

TL;DR

This paper conducts a Hamiltonian analysis of spatially covariant gravity theories, revealing the nature of their constraints and degrees of freedom, and connecting them to a broad class of scalar-tensor theories beyond Horndeski.

Contribution

It shows that in these theories, the lapse function constraints become second class when N enters nonlinearly, leading to an extra scalar degree of freedom beyond general relativity.

Findings

Primary and secondary constraints associated with N are second class.

The theory propagates three degrees of freedom: two tensor and one scalar.

These theories are related to a broad class of scalar-tensor models with higher derivatives.

Abstract

We perform the Hamiltonian constraint analysis for a wide class of gravity theories that are invariant under spatial diffeomorphism. With very general setup, we show that different from the general relativity, the primary and secondary constraints associated with the lapse function $N$ become second class, as long as the lapse function $N$ enters the Hamiltonian nonlinearly. This fact implies that there are three degrees of freedom are propagating, of which two correspond to the usual tensor type transverse and traceless gravitons, and one is the scalar type graviton. By restoring the full spacetime diffeomorphism using the St\"{u}ckelberg trick, this type of spatially covariant gravity theories corresponds to a large class of single field scalar-tensor theories that possess higher order derivatives in the equations of motion, and thus is beyond the scope of the Horndeski theory.