Spatially covariant gravity with a dynamic lapse function

Understand

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).