Stability, ghost, and strong coupling in nonrelativistic general covariant theory of gravity with $\lambda \not=1$

In this paper, we investigate three important issues: stability, ghost and strong coupling, in the Horava-Melby-Thompson setup of the Horava-Lifshitz theory with $\lambda \not= 1$, generalized recently by da Silva. We first develop the general linear scalar perturbations of the Friedmann-Robertson-Walker (FRW) universe with arbitrary spatial curvature, and find that an immediate by-product of the setup is that, in all the inflationary models described by a scalar field, the FRW universe is necessarily flat. Applying them to the case of the Minkowski background, we find that it is stable, and, similar to the case $\lambda = 1$, the spin-0 graviton is eliminated. The vector perturbations vanish identically in the Minkowski background. Thus, similar to general relativity, a free gravitational field in this setup is completely described by a spin-2 massless graviton even with $\lambda \not= 1$. We also study the ghost problem in the FRW background, and find explicitly the ghost-free conditions. To study the strong coupling problem, we consider two different kinds of spacetimes all with the presence of matter, one is cosmological and the one is static. We find that the coupling becomes strong for a process with energy higher than $M_{pl} |c_{\psi}|^{5/2}$ in the flat FRW background, and $M_{pl}|c_{\psi}|^{3}$ in a static weak gravitational field, where $|c_{\psi}| \equiv |(1-\lambda)/(3 \lambda -1)|^{1/2}$.