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 a 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 3 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\"uckelberg trick, this type of spatially covariant gravity theory 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.