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

TL;DR

This paper investigates graviton propagation in deformed Hořava-Lifshitz gravity without the projectability condition, revealing the degrees of freedom and the absence of vDVZ discontinuity at $
$.

Contribution

It analyzes scalar, vector, and tensor modes in the model, clarifies the degrees of freedom, and shows the vDVZ discontinuity is absent for $
$ with Lorentz-violating mass terms.

Findings

Scalar propagation matches general relativity for $
$

Massive gravitons have five degrees of freedom with Lorentz-violating mass terms

vDVZ discontinuity is absent at $
$ in the massless limit

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.