Spatially covariant gravity with velocity of the lapse function: the Hamiltonian analysis

TL;DR

This paper performs a Hamiltonian analysis of spatially covariant gravity theories with kinetic terms for the lapse function, revealing that additional consistency conditions are necessary to eliminate unwanted scalar modes.

Contribution

It identifies the insufficiency of degeneracy conditions alone and introduces a new consistency condition to ensure the removal of scalar modes in these theories.

Findings

Degeneracy condition alone does not eliminate scalar mode.

A new consistency condition is required for the secondary constraint.

Explicit example demonstrating the necessity of the consistency condition.

Abstract

We investigate a large class of gravity theories that respect spatial covariance, and involve kinetic terms for both the spatial metric and the lapse function. Generally such kind of theories propagate four degrees of freedom, one of which is an unwanted scalar mode. Through a detailed Hamiltonian analysis, we find that the condition requiring the kinetic terms to be degenerate is not sufficient to evade the unwanted scalar mode in general. This is because the primary constraint due to the degeneracy condition does not necessarily induce a secondary constraint, if the mixing terms between temporal and spatial derivatives are present. In this case, the second condition that we dub as the consistency condition must be imposed in order to ensure the existence of the secondary constraint and thus to remove the unwanted mode. We also show how our formalism works through an explicit example, in which the degeneracy condition is not sufficient and thus the consistency condition must be imposed.