Jordan and Einstein frames

The Lagrangian in scalar-tensor theory can be expressed in the Jordan frame in which the scalar field or some function of it multiplies the Ricci scalar, or in the Einstein frame in which Ricci scalar is not multiplied by the scalar field. There exist various transformations between these frames. Despite the fact that these frames have been around for some time there is currently heated debate about whether either, both, or neither frame is a 'physical' frame which can be compared to observations and experiment.

If we perform the Weyl rescaling $\tilde{g}_{\mu\nu}=\Phi^{-2/(d-2)} g_{\mu\nu}$, then the Riemann and Ricci tensors are modified.

$\sqrt{-\tilde{g}}=\Phi^{-d/(d-2)}\sqrt{-g}$

$\tilde{R}=\Phi^{2/(d-2)}\left[ R + \frac{6}{d-2}\frac{\Box \Phi}{\Phi} -\frac{6(d-1)}{(d-2)^2}\left(\frac{\nabla\Phi}{\Phi}\right)^2 \right]$

$\int d^dx \sqrt{-\tilde{g}} \Phi \tilde{R} =\int d^dx \sqrt{-g} \left[ R - \frac{6}{(d-2)^2}\left( \nabla\left(\ln \Phi \right) \right)^2\right]$

References

• Valerio Faraoni, Edgard Gunzig, Pasquale Nardone, Conformal transformations in classical gravitational theories and in cosmology, Fundam. Cosm. Phys. 20(1999):121, arXiv:gr-qc/9811047.
• Eanna E. Flanagan, The conformal frame freedom in theories of gravitation, Class. Q. Grav. 21(2004):3817, arXiv:gr-qc/0403063.