A Schrodinger approach to Newton-Cartan and Horava-Lifshitz gravities

We define a 'non-relativistic conformal method', based on a Schrodinger algebra with critical exponent z = 2, as the non-relativistic version of the relativistic conformal method. An important ingredient of this method is the occurrence of a complex compensating scalar field that transforms under both scale and central charge transformations. We apply this non-relativistic method to derive the curved space Newton-Cartan gravity equations of motion with twistless torsion. Moreover, we reproduce z = 2 Horava-Lifshitz gravity by classifying all possible Schrodinger invariant scalar field theories of a complex scalar up to second order in time derivatives.