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 as follows. :\sqrt{-\tilde{g}}=\Phi^{-d/(d-2)}\sqrt{-g} :\tilde{R}=\Phi^{2/(d-2)}\left[ R + \frac{2(d-1)}{d-2}\frac{\Box \Phi}{\Phi} -\frac{3(d-1)}{(d-2)}\left(\frac{\nabla\Phi}{\Phi}\right)^2 \right] As an example consider the transformation of a simple
Scalar-tensor action with an arbitrary set of matter fields \psi_\mathrm{m} coupled minimally to the curved background :S = \int d^dx \sqrt{-\tilde{g}} \Phi \tilde{R} + S_\mathrm{m}[\tilde{g}_{\mu \nu},\psi_\mathrm{m}] =\int d^dx \sqrt{-g} \left[ R + \frac{2(d-1)}{d-2}\frac{\Box \Phi}{\Phi} - \frac{3(d-1)}{(d-2)}\left( \nabla\left(\ln \Phi \right) \right)^2\right] + S_\mathrm{m}[\Phi^{-2/(d-2)} g_{\mu\nu},\psi_\mathrm{m}] The tilde fields then correspond to quantities in the Jordan frame and the fields without the tilde correspond to fields in the Einstein frame. See that the matter action S_\mathrm{m} changes only in the rescaling of the metric. The Jordan and Einstein frames are constructed to render certain parts of physical equations simpler which also gives the frames and the fields appearing in them particular physical interpretations. For instance, in the Einstein frame, the equations for the gravitational field will be of the form :R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu}= \mathrm{other \; fields}\,. I.e., they can be interpreted as the usual
Einstein equations with particular sources on the right-hand side. Similarly, in the
Newtonian limit one would recover the Poisson equation for the
Newtonian potential with separate source terms. However, by transforming to the Einstein frame the matter fields are now coupled not only to the background but also to the field \Phi which now acts as an effective potential. Specifically, an isolated test particle will experience a universal four-acceleration :a^\mu= \frac{-1}{d-2} \frac{\Phi_{,\nu}}{\Phi}(g^{\mu \nu} + u^\mu u^\nu), where u^\mu is the particle
four-velocity. I.e., no particle will be in free-fall in the Einstein frame. On the other hand, in the Jordan frame, all the matter fields \psi_\mathrm{m} are coupled minimally to \tilde{g}_{\mu \nu} and isolated test particles will move on geodesics with respect to the metric \tilde{g}_{\mu \nu}. This means that if we were to reconstruct the
Riemann curvature tensor by measurements of
geodesic deviation, we would in fact obtain the curvature tensor in the Jordan frame. When, on the other hand, we deduce on the presence of matter sources from gravitational lensing from the usual relativistic theory, we obtain the distribution of the matter sources in the sense of the Einstein frame. ==Models==