Hamiltonian analysis of non-projectable modified F(R)F(R) Hořava–Lifshitz gravity

We study a version of the recently proposed modified F(R)F(R) Hořava–Lifshitz gravity that abandons the projectability condition of the lapse variable. We discovered that the projectable version of this theory has a consistent Hamiltonian structure, and that the theory has interesting cosmological solutions which can describe the eras of accelerated expansion of the universe in a unified manner. The usual Hořava–Lifshitz gravity is a special case of our theory. Hamiltonian analysis of the non-projectable theory, however, shows that this theory has serious problems. These problems are compared with those found in the original Hořava–Lifshitz gravity. A general observation on the structure of the Poisson bracket of Hamiltonian constraints in all theories of the Hořava–Lifshitz type is made: in the resulting tertiary constraint the highest order spatial derivative of the lapse N   is always of uneven order. Since the vanishing of the lapse (N=0N=0) is required by the preservation of the Hamiltonian constraints under time evolution, we conclude that the non-projectable version of the theory is physically inconsistent.