An Alternative Approach for General Covariant Horava-Lifshitz Gravity and Matter Coupling

Recently, Horava and Melby-Thompson proposed in arXiv:1007.2410 a nonrelativistic gravity theory with extended gauge symmetry that is free of the spin-0 graviton. We propose a minimal substitution recipe to implement this extended gauge symmetry which reproduce the results obtained by them. Our prescription has the advantage of being manifestly gauge invariant and immediately generalizable to other fields, like matter. We briefly discuss the coupling of gravity with scalar and vector fields found by our method. We show also that the extended gauge invariance in gravity does not force the value of $\lambda$ to be $\lambda=1$ as originally claimed. However, the spin-0 graviton is eliminated even for general $\lambda$.