Conformal transformations single out a unique measure of distance

Abstract

The conformal equivalence theorem states that higher-order gravity theories derived from a Lagrangian which is an analytic function of the scalar curvature are conformally equivalent to general relativity plus a scalar-field matter source with a particular self-interaction effective potential. We discuss the question of physical reality of the two metrics associated with the conformal transformation in the presence of matter fields. We find that of the two conformally related metrics only one can be chosen as the real, physical one. In particular, if the scalar field generated by the conformal transformation satisfies ${\ensuremath{\nabla}}_{a}{\ensuremath{\nabla}}_{b}\ensuremath{\varphi}=0$ then the original metric of the higher-order gravity theory cannot represent a real physical measure of distance and the true metric is that of the system: general relativity plus scalar field. This situation can be realized for a wide class of manifolds.