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

Abstract

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 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 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 backwards. We study energetics for asymptotically flat solutions for those Lagrangians which admit conformal rescaling to 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 + 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.

New constraints on parametrised modified gravity from correlations of the CMB with large scale structure

“…(3), corresponding to the case of a universal fixed coupling α i = 2/3 φ, i.e. β 1 = 4/3 [46]. Moreover, f (R) models that closely mimic ΛCDM correspond to s ∼ 4 [18].…”

Backreaction mechanism in multifluid and extended cosmologies

“…In addition, the relation between averaged quantities in the two frames remains unclear since the domain volumes in different frames don't have a clear correspondence either. On the other hand, besides the mathematical tool that the conformal transformation may provide, the physical meaning of both frames has to be analysed carefully, where the positivity of the energy and the behaviour of the ground state may play a crucial role to discriminate between frames, as pointed out in Refs [51,52].…”

“…Moreover, as pointed out in Refs. [51,52], also the physical meaning of each frame can not be easily established. An analysis of the ground state and the positivity of the energy in both frames is an important tool to determine whether the frames, and in particular the Jordan one, are well defined.…”

“…In this case observers would follow geodesics in the Einstein frame and consequently avoid the appearance of a non-zero acceleration (see Ref. [52]). Nevertheless, in the latter case, the observers in the Jordan frame will not follow geodesics, since a fifth force is induced by the coupling between matter and the scalar field.…”

Conformal transformations and accelerated cosmologies

“…If we assume that only the Einstein frame is directly observable, is the Jordan frame anything more than a mathematical device? There is much debate surrounding these questions (see, e.g., [13], [14], and [15]). Different authors interpret conformal transformations in different ways; the debate often revolves around the question: which of the frames is "physical"?…”