Jordan and Einstein frames

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

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

${\displaystyle {\tilde {R}}=\Phi ^{2/(d-2)}\left[R+{\frac {2(d-1)}{d-2}}{\frac {\Box \Phi }{\Phi }}-{\frac {3(d-1)}{(d-2)}}\left({\frac {\nabla \Phi }{\Phi }}\right)^{2}\right]}$

As an example consider the transformation of a simple Scalar-tensor action with an arbitrary set of matter fields ${\displaystyle \psi _{\mathrm {m} }}$ coupled minimally to the curved background

${\displaystyle S=\int d^{d}x{\sqrt {-{\tilde {g}}}}\Phi {\tilde {R}}+S_{\mathrm {m} }[{\tilde {g}}_{\mu \nu },\psi _{\mathrm {m} }]=\int d^{d}x{\sqrt {-g}}\left[R+{\frac {2(d-1)}{d-2}}{\frac {\Box \Phi }{\Phi }}-{\frac {3(d-1)}{(d-2)}}\left(\nabla \left(\ln \Phi \right)\right)^{2}\right]+S_{\mathrm {m} }[\Phi ^{-2/(d-2)}g_{\mu \nu },\psi _{\mathrm {m} }]}$

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 ${\displaystyle S_{\mathrm {m} }}$ 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

${\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }=\mathrm {other\;fields} \,.}$

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 in the Einstein frame the matter fields are now not coupled simply to the background but also to the field ${\displaystyle \Phi }$ which now acts as an effective potential. Specifically, an isolated test particle will experience a universal four-acceleration

${\displaystyle a^{\mu }={\frac {-1}{d-2}}{\frac {\Phi _{,\nu }}{\Phi }}(g^{\mu \nu }+u^{\mu }u^{\nu }),}$

where ${\displaystyle u^{\mu }}$ 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 ${\displaystyle \psi _{\mathrm {m} }}$ are coupled minimally to ${\displaystyle {\tilde {g}}_{\mu \nu }}$ and isolated test particles will move on geodesics with respect to the metric ${\displaystyle {\tilde {g}}_{\mu \nu }}$. 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.

Jordan frame gravity can be used to calculate type IV singular bouncing cosmological evolution, to derive the type IV singularity.[1]

• Albert Einstein
• Pascual Jordan

• 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.