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On the canonical equivalence between Jordan and Einstein frames

Gabriele Gionti S. J., Matteo Galaverni

Abstract

A longstanding issue is the classical equivalence between the Jordan and the Einstein frames, which is considered just a field redefinition of the metric tensor and the scalar field. In this work, based on the previous result that the Hamiltonian transformations from the Jordan to the Einstein frame are not canonical on the extended phase space, we study the possibility of the existence of canonical transformations. We show that on the reduced phase space -- defined by suitable gauge fixing of the lapse and shifts functions -- these transformations are Hamiltonian canonical. Poisson brackets are replaced by Dirac's brackets following the Bergman-Dirac's procedure. The Hamiltonian canonical transformations map solutions of the equations of motion in the Jordan frame into solutions of the equations of motion in the Einstein frame.

On the canonical equivalence between Jordan and Einstein frames

Abstract

A longstanding issue is the classical equivalence between the Jordan and the Einstein frames, which is considered just a field redefinition of the metric tensor and the scalar field. In this work, based on the previous result that the Hamiltonian transformations from the Jordan to the Einstein frame are not canonical on the extended phase space, we study the possibility of the existence of canonical transformations. We show that on the reduced phase space -- defined by suitable gauge fixing of the lapse and shifts functions -- these transformations are Hamiltonian canonical. Poisson brackets are replaced by Dirac's brackets following the Bergman-Dirac's procedure. The Hamiltonian canonical transformations map solutions of the equations of motion in the Jordan frame into solutions of the equations of motion in the Einstein frame.
Paper Structure (16 sections, 190 equations, 2 figures)

This paper contains 16 sections, 190 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Hamiltonian Weyl (conformal) transformations in FLRW Universe
  3. Gauge fixing in the FLRW case
  4. Jordan frame
  5. Einstein frame
  6. Hamiltonian Weyl (conformal) transformations in the general case
  7. Gauge fixing in the general case
  8. Jordan frame
  9. Einstein frame
  10. Discussion and Conclusions
  11. Brans-Dicke theory in flat FLRW universe for $\omega = -3/2$
  12. Gauge fixing in the JF
  13. Gauge fixing in the EF
  14. Brans-Dicke theory in ADM spacetime for $\omega=-3/2$
  15. Gauge fixing in the JF
  16. ...and 1 more sections

Figures (2)

  • Figure 1: On the extended phase space we transform ${H}_{T}$ (the total Hamiltonian in the JF) into $\widetilde{H}_{T}$ (the total Hamiltonian in the EF) using the relations between variables and conjugate momenta in the two frames, see \ref{['JFEFtrans0']} and \ref{['JFEFtrans']}. It is not possible to pass from the equations of motion in the JF to the equations of motion in EF (and vice-versa) simply using the relations \ref{['JFEFtrans0']} and \ref{['JFEFtrans']}, for more details see Galaverni:2021jcy.
  • Figure 2: On the sub-manifold defined by the vanishing of the second class constraints the equations of motion in the EF \ref{['adotEF2']}-\ref{['piphidotEF2']} can be mapped into the equations of motion in the JF \ref{['adotJF2']}-\ref{['piphidotJF2']}, and vice-versa, using the transformations \ref{['JFEFtrans']} together with the Hamiltonian constraint $H\approx0$ defined in Eq. \ref{['H:JF:neq']}. This is a clear consequence of the canonicity of the Weyl (conformal) transformation between JF and EF on the sub-manifold (reduced phase space) defined with the gauge fixing conditions.