Hamiltonian analysis of spatially covariant gravity

We perform the Hamiltonian constraint analysis for a wide class of gravity theories that are invariant under spatial diffeomorphism. With very general setup, we show that different from the general relativity, the primary and secondary constraints associated with the lapse function $N$ become second class, as long as the lapse function $N$ enters the Hamiltonian nonlinearly. This fact implies that there are three degrees of freedom are propagating, of which two correspond to the usual tensor type transverse and traceless gravitons, and one is the scalar type graviton. By restoring the full spacetime diffeomorphism using the St\"{u}ckelberg trick, this type of spatially covariant gravity theories corresponds to a large class of single field scalar-tensor theories that possess higher order derivatives in the equations of motion, and thus is beyond the scope of the Horndeski theory.