In the context of Scalar-Tensor theories of gravity (for example in Brans-Dicke) what is the difference between gravitational and matter scalar Fields?
My doubt comes from "The scalar-tensor Theory of gravitation" of Y. Fujii and K. Maeda (pag. 68, near formula 3.34).
Clearly, a scalar field is a gravitational one in the Jordan frame if it's coupled to $R$ into the action. With a conformal transformation we can go to the Einstein frame, in which the (redefined) field is now coupled to matter. I guess that this is still a gravitational scalar field, because is derived from the first field.
So, it's correct to say that a field is a matter field if it's only in the matter part of the action in the Jordan frame? I think that this can be true, since Jordan frame is often called the "physical" one.
And in the Einstein frame? It seems to me that every scalar field coupled to matter can be transformed in a non-minimally coupled field into the Jordan frame.