$z=3$ Lifshitz-Hořava model and Fermi-point scenario of emergent gravity

Recently Hořava proposed a model for gravity which is described by the Einstein action in the infrared, but lacks the Lorentz invariance in the high-energy region where it experiences the anisotropic scaling. We test this proposal using two condensed matter examples of emergent gravity: acoustic gravity and gravity emerging in the fermionic systems with Fermi points. We suggest that quantum hydrodynamics, which together with the quantum gravity is the non-renormalizable theory, may exhibit the anisotropic scaling in agreement with the proposal. The Fermi point scenario of emergent general relativity demonstrates that under general conditions, the infrared Einstein action may be distorted, i.e. the Hořava parameter $\lambda$ is not necessarily equal 1 even in the low energy limit. The consistent theory requires special hierarchy of the ultra-violet energy scales and the fine-tuning mechanism for the Newton constant.