Schrödinger-like Dilaton Gravity

We investigate possibilities for a Schrödinger-like gravity with the dynamical critical exponent z = 2, where the action only contains the first-order time derivative. The Horava gravity always admits such a relevant deformation because the full (d + 1) dimensional diffeomorphism of the Einstein gravity is replaced by the foliation preserving diffeomorphism. The dynamics is locally trivial or topological in the pure gravity case, but we can construct a dynamical field theory with a z = 2 dispersion relation by introducing a dilaton degree of freedom. Our model provides a classical starting point for the possible quantum dilaton gravity which may be applied to a membrane quantization.