Spatially covariant gravity with a dynamic lapse function

TL;DR

This paper explores spatially covariant gravity with a dynamic lapse function, establishing conditions for two physical degrees of freedom and constructing quadratic actions that extend general relativity.

Contribution

It identifies necessary and sufficient conditions for two DoF in such theories and constructs new quadratic actions, including transformations from GR.

Findings

Two conditions ensure two physical degrees of freedom.

Certain quadratic actions cannot recover GR due to curvature restrictions.

Some quadratic actions are related to GR via disformal transformations.

Abstract

In the framework of spatially covariant gravity, it is natural to extend a gravitational theory by putting the lapse function $N$ and the spatial metric $h_{ij}$ on an equal footing. We find two sufficient and necessary conditions for ensuring two physical degrees of freedom (DoF) for the theory with the lapse function being dynamical by Hamiltonian analysis. A class of quadratic actions with only two DoF is constructed. In the case that the coupling functions depend on $N$ only, we find that the spatial curvature term cannot enter the Lagrangian and thus this theory possesses no wave solution and cannot recover general relativity (GR). In the case that the coupling functions depend on the spatial derivatives of $N$, we perform a spatially conformal transformation on a class of quadratic actions with nondynamical lapse function to obtain a class of quadratic actions with $\dot{N}$. We confirm this theory has two DoF by checking the two sufficient and necessary conditions. Besides, we find that a class of quadratic actions with two DoF can be transformed from GR by disformal transformation.