Equivalent frames in Brans-Dicke theory

We discuss the physical equivalence between the Einstein and Jordan frames in Brans-Dicke theory. The inequivalence of conformal transformed theories is clarified with the help of an old equivalence theorem of Chrisholm’s. 04.20. Fy, 04.50. +h Typeset using REVTEX 1 Brans-Dicke theory is a natural generalization of Einstein theory. In Brans-Dicke theory, the effective gravitational constantGeff = φ , varies as the Brans-Dicke field φ evolves. The evolution of the scalar field slows down the expansion rate of the universe during inflation, and allows nucleation of bubbles to end the inflationary era. There are two conformally related frames, the Einstein frame and the Jordan frame, in Brans-Dicke theory. In the literature, people do not agree with each other about the equivalence of the two frames. Some people think that the two frames are physically equivalent and some people do not think that they are physically equivalent. For example, some people considered the inflationary models in Einstein frame in order to solve equations easily, but they analyzed their final results in Jordan frame because most people insist that the Jordan frame is the physical frame to keep the equivalence principle. In fact, the equivalence principle can be kept in Einstein frame if we use Einstein frame as the physical frame. A dilaton field appears in Kaluza-Klein reduction and string theory also. In string theory, we have the conformally related string frame and Einstein frame. As we all know, the string frame is equivalent to the Einstein frame. How could that the string frame is equivalent to the Einstein frame and the Jordan frame is not equivalent to the Einstein frame? In this letter, we will use the equivalence theorem to address this problem. In general, we can ask if the physics remains the same under an arbitrary change of field variables. Let us first look at the equivalence theorem [1] [2] [3]. Theorem: Let the Lagrangian be known in terms of a set of field variables φ, L[φ]. If one expresses the field variables φ as nonlinear but local functions of another set of field variables φ (one may think of φ as a scalar field for simplicity), φ = f [φ], f [0] = 0, f ′[φ] 6= 0, (1) one can write down the Lagrangian as L[φ] = L[f(φ)] = Lt[φ]. The on-mass shell S matrices calculated with L[φ] and Lt[φ] are identical (in making the comparison it may be necessary to introduce appropriate wave-function renormalizations). In other words, both the fields φ and φ can be used to describe the same physics.