Hamiltonian analysis of nonprojectable Ho\v{r}ava-Lifshitz gravity with
  $U(1)$ symmetry
  We study the nature of constraints and count the number of degrees of freedom
in the nonprojectable version of the $U(1)$ extension of Ho\v{r}ava-Lifshitz
gravity, using the standard method of Hamiltonian analysis in the classical
field theory. This makes it possible for us to investigate the condition under
which the scalar graviton is absent at a fully nonlinear level. We show that
the scalar graviton does not exist at the classical level if and only if two
specific coupling constants are exactly zero. The operators corresponding to
these two coupling constants are marginal for any values of the dynamical
critical exponent of the Lifshitz scaling and thus should be generated by
quantum corrections even if they are eliminated from the bare action. We thus
conclude that the theory in general contains the scalar graviton.
