Hamiltonian analysis of nonprojectable Ho\v{r}ava-Lifshitz gravity with $U(1)$ symmetry

摘要

We study the nature of constraints and count the number of degrees of freedomin the nonprojectable version of the $U(1)$ extension of Ho\v{r}ava-Lifshitzgravity, using the standard method of Hamiltonian analysis in the classicalfield theory. This makes it possible for us to investigate the condition underwhich the scalar graviton is absent at a fully nonlinear level. We show thatthe scalar graviton does not exist at the classical level if and only if twospecific coupling constants are exactly zero. The operators corresponding tothese two coupling constants are marginal for any values of the dynamicalcritical exponent of the Lifshitz scaling and thus should be generated byquantum corrections even if they are eliminated from the bare action. We thusconclude that the theory in general contains the scalar graviton.