Spatially covariant gravity with a dynamic lapse function

In the framework of spatially covariant gravity (SCG), 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 freedoms (DoFs) for the theory with the lapse function being dynamical by Hamiltonian analysis. A class of quadratic actions with only two DoFs 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 non-dynamical lapse function to obtain a class of quadratic actions with $\dot{N}$. We confirm this theory has two DoFs by checking the two conditions. Besides, we find that a class of quadratic actions with two DoFs can be transformed from GR by disformal transformation.