Lorentz breaking terms from Einstein gravity

We construct an action which is invariant under the foliation preserving diffeomorphism from the Einstein Hilbert action. Starting from the Einstein Hilbert action, we introduce the gauge invariance under the anisotropic rescaling by using the St\"uckelberg method. We then introduce the gauge field corresponds to the anisotropic rescaling, and the St\"uckelberg field turns out to be the Nambu-Goldstone boson. The Nambu-Goldstone boson, however, is not completely eaten by the massive gauge field and the Nambu-Goldstone boson can be integrated out from the action. Then the resultant action is a Horava-Lifshitz type action which contains $R^3$ term.