Space- vs Time-dependence in taming the infrared instability of projectable Ho\v{r}ava Gravity
Minkowski spacetime exhibits infrared instability in projectable Ho\v rava gravity in (3+1) dimensions. To be phenomenologically viable, the instability should be either hidden by other time-dependent processes such as the Hubble expansion of the universe and the Jeans instability, or evolve into another static solution with low average curvature. While the former scenario leads to a phenomenological constraint on the infrared properties of the renormalization group flow, this paper explores the latter possibility. We study if the presence of higher derivative terms in the action can lead to existence of static, inhomogeneous (quasi-) periodic solutions with planar symmetry, similar to modulated phases in magnetic materials. We find that such solutions do not exist. In doing so, we classify all static homogeneous and isotropic solutions and solutions with planar symmetry. We provide arguments that none of them can serve as an endpoint for the evolution of the Minkowski instability. This motivates further study of the scenario where the instability is concealed by time evolution.