Spatially covariant gravity with two degrees of freedom in the presence of an auxiliary scalar field: Hamiltonian analysis

TL;DR

This paper performs a Hamiltonian analysis of a class of spatially covariant gravity theories with auxiliary scalar fields, identifying conditions to reduce the propagating degrees of freedom from three to two, consistent with gravitational wave modes.

Contribution

It provides a Hamiltonian constraint analysis for spatially covariant gravity with auxiliary scalars, establishing conditions to eliminate scalar degrees of freedom and confirming previous perturbative results.

Findings

Two conditions are necessary to eliminate the scalar degree of freedom.

The second condition's effect depends on the Dirac matrix classification.

Application to a polynomial Lagrangian confirms the theoretical conditions.

Abstract

A class of gravity theories respecting spatial covariance and in the presence of non-dynamical auxiliary scalar fields with only spatial derivatives is investigated. Generally, without higher temporal derivatives in the metric sector, there are 3 degrees of freedom (DOFs) propagating due to the breaking of general covariance. Through a Hamiltonian constraint analysis, we examine the conditions to eliminate the scalar DOF such that only 2 DOFs, which correspond the tensorial gravitational waves in a homogeneous and isotropic background, are propagating. We find that two conditions are needed, each of which can eliminate half degree of freedom. The second condition can be further classified into two cases according to its effect on the Dirac matrix. We also apply the formal conditions to a polynomial-type Lagrangian as a concrete example, in which all the monomials are spatially covariant scalars containing two derivatives. Our results are consistent with the previous analysis based on the perturbative method.