Massive graviton propagation of the deformed Ho\v{r}ava-Lifshitz gravity without projectability condition

Abstract

We study graviton propagations of scalar, vector, and tensor modes in the deformed Ho\v{r}ava-Lifshitz gravity ($\lambda R$-model) without projectability condition. The quadratic Lagrangian is invariant under diffeomorphism only for $\lambda=1$ case, which contradicts to the fact that $\lambda$ is irrelevant to a consistent Hamiltonian approach to the $\lambda R$ model. In this case, as far as scalar propagations are concerned, there is no essential difference between deformed Ho\v{r}ava-Lifshitz gravity ($\lambda R$-model) and general relativity. This implies that there are two degrees of freedom for a massless graviton without Ho\v{r}ava scalar, and five degrees of freedom appear for a massive graviton when introducing Lorentz-violating and Fierz-Pauli mass terms. Finally, it is shown that for $\lambda=1$, the vDVZ discontinuity is absent in the massless limit of Lorentz-violating mass terms by considering external source terms.