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.

Equations and physical interpretationEdit

If we perform the Weyl rescaling  , then the Riemann and Ricci tensors are modified as follows.


As an example consider the transformation of a simple Scalar-tensor action with an arbitrary set of matter fields   coupled minimally to the curved background


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   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


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   which now acts as an effective potential. Specifically, an isolated test particle will experience a universal four-acceleration


where   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   are coupled minimally to   and isolated test particles will move on geodesics with respect to the metric  . 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]

See alsoEdit


  1. ^ S.D. Odintsov, V.K. Oikonomou (27 June 2015). "Bouncing cosmology with future singularity from modified gravity". Physical Review D. 92 (2): 024016. arXiv:1504.06866. Bibcode:2015PhRvD..92b4016O. doi:10.1103/PhysRevD.92.024016.
  • 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.