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    Schr\"odinger-like Dilaton Gravity

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    We investigate possibilities for a Schr\"odinger-like gravity with the dynamical critical exponent z=2z=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)(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=2z=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
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