  We investigate a large class of gravity theories that respect spatial
covariance, and involve kinetic terms for both the spatial metric and the lapse
function. Generally such kind of theories propagate four degrees of freedom,
one of which is an unwanted scalar mode. Through a detailed Hamiltonian
analysis, we find that the condition requiring the kinetic terms to be
degenerate is not sufficient to evade the unwanted scalar mode in general. This
is because the primary constraint due to the degeneracy condition does not
necessarily induce a secondary constraint, if the mixing terms between temporal
and spatial derivatives are present. In this case, the second condition that we
dub as the consistency condition must be imposed in order to ensure the
existence of the secondary constraint and thus to remove the unwanted mode. We
also show how our formalism works through an explicit example, in which the
degeneracy condition is not sufficient and thus the consistency condition must
be imposed.
