Physical equivalence between nonlinear gravity theories and a general-relativistic self-gravitating scalar field

We argue that in a nonlinear gravity theory (the Lagrangian being an arbitrary function of the curvature scalar R), which according to well-known results is dynamically equivalent to a self-gravitating scalar field in general relativity, the true physical variables are exactly those which describe the equivalent general-relativistic model (these variables are known as the Einstein frame). Whenever such variables cannot be defined, there are strong indications that the original theory is unphysical, in the sense that Minkowski space is unstable due to the existence of negative-energy solutions close to it. To this aim we first clarify the global net of relationships between the nonlinear gravity theories, scalar-tensor theories, and general relativity, showing that in a sense these are ‘‘canonically conjugated’’ to each other. We stress that the isomorphisms are in most cases local; in the regions where these are defined, we explicitly show how to map, in the presence of matter, the Jordan frame to the Einstein one and vice versa. We study energetics for asymptotically flat solutions for those Lagrangians which admit conformal rescaling to the Einstein frame in the vicinity of flat space. This is based on the second-order dynamics obtained, without changing the metric, by the use of a Helmholtz Lagrangian. We prove for a large class of these Lagrangians that the ADM energy is positive for solutions close to flat space, and this is determined by the lowest-order terms R+${\mathit{aR}}^2$ in the Lagrangian. The proof of this positive-energy theorem relies on the existence of the Einstein frame, since in the (Helmholtz-)Jordan frame the dominant energy condition does not hold and the field variables are unrelated to the total energy of the system. This is why we regard the Jordan frame as unphysical, while the Einstein frame is physical and reveals the physical contents of the theory. The latter should hence be viewed as physically equivalent to a self-interacting general-relativistic scalar field.