Equivalence of the Einstein and Jordan frames

Abstract

No experiment can measure an absolute scale: every dimensionful quantity has to be compared to some fixed unit scale in order to be measured, and thus only dimensionless quantities are really physical. The Einstein and Jordan frames are related by a conformal transformation of the metric, which amounts to rescaling all length scales. Since the absolute scale cannot be measured, both frames describe the same physics and are equivalent. In this article we make this explicit by rewriting the action in terms of dimensionless variables, which are invariant under a conformal transformation. For definitiveness, we concentrate on the action of Higgs inflation, but the results can easily be generalized. In addition, we show that the action for $f(R)$ gravity, which includes Starobinsky inflation, can be written in a frame-independent form.