Stability, ghost, and strong coupling in nonrelativistic general covariant theory of gravity withλ≠1

Abstract

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 $\ensuremath{\lambda}\ensuremath{\ne}1$, 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 $\ensuremath{\lambda}=1$, 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 $\ensuremath{\lambda}\ensuremath{\ne}1$. 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 ${M}_{\mathrm{pl}}|{c}_{\ensuremath{\psi}}{|}^{5/2}$ in the flat FRW background and ${M}_{\mathrm{pl}}|{c}_{\ensuremath{\psi}}{|}^{3}$ in a static weak gravitational field, where $|{c}_{\ensuremath{\psi}}|\ensuremath{\equiv}|(1\ensuremath{-}\ensuremath{\lambda})/(3\ensuremath{\lambda}\ensuremath{-}1){|}^{1/2}$.