Stability, ghost, and strong coupling in nonrelativistic general covariant theory of gravity with $\lambda \ne 1$

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 $\lambda \ne 1$, 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 $\lambda \ne 1$. 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 other is static. We find that the coupling becomes strong for a process with energy higher than $M_{\mathrm{pl}}|c_{\psi }{|}^{5/2}$ in the flat FRW background and $M_{\mathrm{pl}}|c_{\psi }{|}^3$ in a static weak gravitational field, where $|c_{\psi }|\equiv |(1-\lambda )/(3\lambda -1){|}^{1/2}$.