Spatially covariant gravity with a dynamic lapse function

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