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.
We derive a generic identity which holds for the metric (i.e. variational) energy–momentum tensor under any field transformation in any generally covariant classical Lagrangian field theory. The identity determines the conditions under which a symmetry of the Lagrangian is also a symmetry of the energy–momentum tensor. It turns out that the stress tensor acquires the symmetry if the Lagrangian has the symmetry in a generic curved spacetime. In this sense, a field theory in flat spacetime is not self-contained. When the identity is applied to the gauge invariant spin-2 field in Minkowski space, we obtain an alternative and direct derivation of a known no-go theorem: a linear gauge invariant spin-2 field, which is dynamically equivalent to linearized general relativity, cannot have a gauge invariant metric energy–momentum tensor. This implies that attempts to define the notion of gravitational energy density in terms of the metric energy–momentum tensor in a field-theoretical formulation of gravity must fail.
The conformal ambiguity in the definition of the four-dimensional reduced metric arises in any theory whose higher-dimensional metric contains scalar fields. If d=5 the reduced metric can be uniquely determined by its relation to the total five-dimensional gravitational energy. For d)5 the notion of gravitational energy is as yet undefined and for one scalar field several authors have used various arguments in order to single out one (and the same in all cases) reduced metric. It is shown that a unique choice of the reduced metric (i.e. a unique conformal factor) follows from the positive-energy theorem in four dimensions. For a generic case when the spacetime evolution of the internal homogeneous space is described by n scalar fields, the author proves the existence of an analogous conformal factor, ensuring that the total kinetic energy in four dimensions satisfies the dominant energy condition.
We critically review some concepts underlying current applications of gravity theories with Lagrangians L = f (g µν , R αβµν ) to cosmology to account for the accelerated expansion of the universe. We argue that one cannot reconstruct the function f from astronomical observations either in cosmology or in the solar system. The Robertson-Walker spacetime is so simple and "flexible" that any cosmic evolution may be fitted by infinite number of various Lagrangians. We show on the example of Newton's gravity that one cannot recover the correct equation of motion from its approximate solution. Any gravity theory different from general relativity generates a new cosmological theory and all the successes of the standard cosmological model are lost even if a single solution of the theory well fits the observations. Prior to application of a given gravity theory to cosmology or elsewhere it is necessary to establish its physical contents and viability. This study may be performed by a universal method of Legendre transforming the initial Lagrangian in a Helmholtz Lagrangian. In this formalism Lagrange equations of motion are of second order and are the Einstein field equations with additional massive spin-zero and spin-two fields. All the gravity theories differ only by a form of interaction terms of the two fields and the metric. Initial conditions for the two fields in the gravitational triplet depend on which frame (i.e., the set of dynamical variables) is physical (i.e. matter is minimally coupled in it). This fact and the multiplicity of possible frames obstruct confrontation of solutions to equations of motion with the observational data. A fundamental and easily applicable criterion of viability of any gravity theory is the existence of a stable ground state solution being either Minkowski, de Sitter or anti-de Sitter space. Stability of the ground state is independent of which frame is physical.
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