On the equivalence of Jordan and Einstein frames in scale-invariant gravity

Abstract

In this note we consider the issue of the classical equivalence of scale-invariant gravity in the Einstein and in the Jordan frames. We first consider the simplest example f (R) = R 2 and show explicitly that the equivalence breaks down when dealing with Ricci-flat solutions. We discuss the link with the fact that flat solutions in quadratic gravity have zero energy. We also consider the case of scaleinvariant tensor-scalar gravity and general f (R) theories. We argue that all scale-invariant gravity models have Ricci flat solutions in the Jordan frame that cannot be mapped into the Einstein frame. In particular, the Minkowski metric exists only in the Jordan frame. In this sense, the two frames are not equivalent. * Electronic address: massimiliano.rinaldi@unitn.it

Dynamical equivalence of f(R) gravity in Jordan and Einstein frames

“…Moreover, we will show that, when both frames exist, they are fully equivalent from the dynamical point of view, even though the dynamical system formulation in the two frames are not in general diffeomorphic or, in the dynamical system language, topologically equivalent. Such results could help not only the construction of viable f (R) cosmological models, but also contribute to the still active debate on the physical interpretation of the two frames, see [17][18][19][20][21][22][23][24][25][26][27][28] for other recent references on this issue and many of its implications in different physical contexts. In particular, for the question on the frame equivalence at the quantum level, see [29][30][31].…”

Spherical inhomogeneous solutions of Einstein and scalar-tensor gravity: a map of the land

“…• Rinaldi and generalized Rinaldi solutions (Anabalón et al, 2014;Minamitsuji, 2014;Rinaldi, 2012): Without entering into the details of the derivation, these solutions correspond to q = 0 and β > 0 and are given by…”

“…With the exception of.30 Care must be taken, where the map between conformal frames breaks down (in R 2 gravity, where R = 0(Rinaldi, 2018)). 31 This proof is repeated, but specifically for f (R) gravity instead of general scalar-tensor theory, in(Ravindranath et al, 2018).…”

Scale-invariant inflation with one-loop quantum corrections

“…Nevertheless, it can be argued that the two descriptions should match on-shell in order to have the correct S-matrix elements, see e.g. [35][36][37] (on the equivalence of the two frames in the space of solutions see however [38]). With the pseudo-optimal energy scale choice the Lagrangian in the Einstein frame is…”