Uniqueness of the metric line element in dimensionally reduced theories

Abstract

The conformal ambiguity in the definition of the four-dimensional reduced metric arises in any theory whose higher-dimensional metric contains scalar fields. If d=5 the reduced metric can be uniquely determined by its relation to the total five-dimensional gravitational energy. For d)5 the notion of gravitational energy is as yet undefined and for one scalar field several authors have used various arguments in order to single out one (and the same in all cases) reduced metric. It is shown that a unique choice of the reduced metric (i.e. a unique conformal factor) follows from the positive-energy theorem in four dimensions. For a generic case when the spacetime evolution of the internal homogeneous space is described by n scalar fields, the author proves the existence of an analogous conformal factor, ensuring that the total kinetic energy in four dimensions satisfies the dominant energy condition.