Uniqueness of the metric line element in dimensionally reduced theories

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. We show 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, we prove the existence of an analogous conformal factor, ensuring that the total kinetic energy in four dimensions satisfies the dominant energy condition. (author)