The Lagrangian in scalar-tensor theory can be expressed in the Jordan frame or in the Einstein frame, which are field variables that stress different aspects of the gravitational field equations and the evolution equations of the matter fields.
As a result, in the Einstein frame the field equations for the space-time metric resemble the Einstein equations but test particles do not move on geodesics of the metric.
Christopher Hill and Graham Ross have shown that there exist ``gravitational contact terms" in the Jordan frame, whereby the action is modified by graviton exchange.
This modification leads back to the Einstein frame as the effective theory.
[1] Contact interactions arise in Feynman diagrams when a vertex contains a power of the exchanged momentum,
When the contact term is included results for amplitudes in the Jordan frame will be equivalent to those in the Einstein frame, and results of physical calculations in the Jordan frame that omit the contact terms will generally be incorrect.
As an example consider the transformation of a simple Scalar-tensor action with an arbitrary set of matter fields
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.
Specifically, an isolated test particle will experience a universal four-acceleration where
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.