Massive graviton propagation of the deformed Hořava-Lifshitz gravity without projectability condition

We study graviton propagations of scalar, vector, and tensor modes in the deformed Hořava-Lifshitz gravity ($λR$-model) without projectability condition. The quadratic Lagrangian is invariant under diffeomorphism only for $λ=1$ case, which contradicts to the fact that $λ$ is irrelevant to a consistent Hamiltonian approach to the $λR$ model. In this case, as far as scalar propagations are concerned, there is no essential difference between deformed Hořava-Lifshitz gravity ($λR$-model) and general relativity. This implies that there are two degrees of freedom for a massless graviton without Hořava 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 $λ=1$, the vDVZ discontinuity is absent in the massless limit of Lorentz-violating mass terms by considering external source terms.

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