A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities

We define a ‘non-relativistic conformal method’, based on a schrödinger 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 hořava-lifshitz gravity by classifying all possible schrödinger invariant scalar field theories of a complex scalar up to second order in time derivatives.