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.
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