If one assumes higher dimensions and that dimensional reduction from higher dimensions produces scalar-tensor theory and also that Palatini variation is the correct method of varying scalar-tensor theory then spacetime is nonmetric. Palatini variation of Jordan frame lagrangians gives an equation relating the dilaton to the object of non-metricity and hence the existence of the dilaton implies that the spacetime connection is more general than that given soley by the Christoffel symbol of general relativity. Transferring from Jordan to Einstein frame, which connection, lagrangian, field equations and stress conservation equations occur are discussed: it is found that the Jordan frame has more information, this can be expressed in several ways, the simplest is that the extra information corresponds to the function multiplying the Ricci scalar in the action. The Einstein frame has the advantages that stress conservation implies no currents and that the field equations are easier to work with. This is illustrated by application to Robertson-Walker spacetime.
The most general second-order field equations of bi-scalar-tensor theory in four dimensions
