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. In the Jordan frame the scalar field or some function of it multiplies the Ricci scalar in the Lagrangian and the matter is typically coupled minimally to the metric, whereas in the Einstein frame the Ricci scalar is not multiplied by the scalar field and the matter is coupled non-minimally. 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. On the other hand, in the Jordan frame test particles move on geodesics, but the field equations are very different from Einstein equations. The causal structure in both frames is always equivalent and the frames can be transformed into each other as convenient for the given application. 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. Contact interactions arise in Feynman diagrams when a vertex contains a power of the exchanged momentum, q 2 {\displaystyle q^{2}} , which then cancels against the Feynman propagator, 1 / q 2 {\displaystyle 1/q^{2}} , leading to a point-like interaction. This must be included as part of the effective action of the theory. 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. This implies that the Jordan frame action is misleading, and the Einstein frame is uniquely correct for fully representing the physics.
Equations and physical interpretation If we perform the Weyl rescaling g ~ μ ν = Φ − 2 / ( d − 2 ) g μ ν {\displaystyle {\tilde {g}}_{\mu \nu }=\Phi ^{-2/(d-2)}g_{\mu \nu }} , then the Riemann and Ricci tensors are modified as follows.
− g ~ = Φ − d / ( d − 2 ) − g {\displaystyle {\sqrt {-{\tilde {g}}}}=\Phi ^{-d/(d-2)}{\sqrt {-g}}}
R ~ = Φ 2 / ( d − 2 ) [ R + 2 ( d − 1 ) d − 2 ◻ Φ Φ − 3 ( d − 1 ) ( d − 2 ) ( ∇ Φ Φ ) 2 ] {\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 ψ m {\displaystyle \psi _{\mathrm {m} }} coupled minimally to the curved background
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