Abstract
We analyze the Hamiltonian equivalence between Jordan and Einstein frames considering a mini-superspace model of the flat Friedmann–Lemaître–Robertson–Walker (FLRW) Universe in the Brans–Dicke theory. Hamiltonian equations of motion are derived in the Jordan, Einstein, and anti-gravity (or anti-Newtonian) frames. We show that, when applying the Weyl (conformal) transformations to the equations of motion in the Einstein frame, we did not obtain the equations of motion in the Jordan frame. Vice-versa, we re-obtain the equations of motion in the Jordan frame by applying the anti-gravity inverse transformation to the equations of motion in the anti-gravity frame.
Highlights
The Brans–Dicke theory [1] is a special case of scalar-tensor theory [2]: Z p ω μν S =d x − g φ R − g ∂μ φ∂ν φ − U (φ) φ M √ + 2 d x hφK .Gaetano Luciano, LucianoPetruzziello and Luca SmaldoneReceived: 30 November 2021Accepted: 21 December 2021The equations of motion for the metric tensor gμν are: Rμν − gμν R
In two papers [3,4], we have studied in detail the Hamiltonian theory related to the action (1), where we focused on the issue of canonical transformations
We have introduced Jordan and Einstein frames [2,6,7] that are defined starting from the Brans–Dicke theory and the Hamiltonian transformations between the two frames, both for the case of ω 6= − 32 and ω = − 32 [8]
Summary
The Brans–Dicke theory [1] is a special case of scalar-tensor theory [2]: Z p ω μν. + 2 d x hφK. We have introduced Jordan and Einstein frames [2,6,7] that are defined starting from the Brans–Dicke theory and the Hamiltonian transformations between the two frames, both for the case of ω 6= − 32 and ω = − 32 [8]. The solutions of the equations of motion in one frame are not necessarily the solutions of the equations of motion in the other frame We show this in an even clearer way by considering, in the Brans–Dicke theory, a flat FLRW minisuperspace model. We derive the Hamiltonian functions and the equations of motion in the Jordan frame and in the Einstein frame. We notice that only if we employ the anti-gravity (or anti-Newtonian) transformations that the previous procedure reproduces, in the Jordan frame, do we derive the same equations of motion as those from the original Hamiltonian.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
