Jordan and Einstein frames Hamiltonian analysis for FLRW Brans-Dicke theory

TL;DR

This paper investigates the Hamiltonian equivalence between Jordan and Einstein frames in Brans-Dicke theory within a flat FLRW universe, revealing that conformal transformations do not preserve equations of motion across frames.

Contribution

It provides a detailed Hamiltonian analysis showing the non-equivalence of equations of motion under Weyl transformations between the frames.

Findings

Weyl transformations do not map equations of motion between Einstein and Jordan frames.

Equations of motion in the Jordan frame are recovered via anti-gravity inverse transformations.

Hamiltonian equations are explicitly derived in multiple frames.

Abstract

We analyze Hamiltonian equivalence between Jordan and Einstein frames considering a mini-superspace model of flat Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe in Brans-Dicke theory. Hamiltonian equations of motion are derived in the Jordan, Einstein, and in the anti-gravity (or anti-Newtonian) frames. We show that applying the Weyl (conformal) transformations to the equations of motion in the Einstein frame we did not get the equations of motion in the Jordan frame. Vice-versa, we re-obtain the equations of motion in the Jordan frame applying the anti-gravity inverse transformation to the equation of motion in the anti-gravity frame.