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.

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Among the so-called 'non-linear' (purely metric) Lagrangians for the gravitational field, those which depend in a quadratic way on the components of the Riemann tensor have been given particular consideration by many authors. In this paper, the authors deal with the most general quadratic Lagrangian depending on the full Riemann tensor, in arbitrary dimension; instead of considering the corresponding fourth-order Euler-Lagrange equations, they investigate an equivalent set of second-order quasilinear equations which are obtained by (a suitably generalised) Legendre transformation. In this framework, they compare this class of theories with those depending on the Ricci tensor only, showing that the Weyl tensor dependence breaks the equivalence with general relativity, but the new auxiliary field arising in this case has no dynamical term. The degeneracy occurring for a suitable choice of the parameters in the Lagrangian is widely discussed, and some effects of a non-minimal coupling with an external scalar field are also described.

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.

Earlier results of Higgs (1959), Stelle (1978) and Whitt (1984) on the dynamical equivalence between Einstein's theory and a class of quadratic theories of gravitation are analysed in view of a more general result, implying the same conclusion for a much larger class of theories (essentially all those which depend arbitrarily on the Ricci tensor). It is shown that all previously known cases of conformal equivalence follow from the prescription of a general Legendre transform, which is in fact suggested by an earlier idea of Einstein (1923) and Eddington (1924). The extension to cover also the dependence on Weyl's tensor (1950) is shortly discussed.

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In reply to some observations by Brans the authors comment on the notion of 'physical metric' in the framework of equivalent metric theories of gravitation. They add some relevant remarks on the very concept of dynamical equivalence.

We show that the direct sum of n copies of a Lie algebra is endowed with a sequence of affine Lie-Poisson brackets, which are pairwise compatible and define a multi-Hamiltonian structure; to this structure one can associate a recursion operator and a Kac-Moody algebra of Hamiltonian vector fields. If the initial Lie algebra is taken to be an associative algebra of differential operators, a suitable family of Hamiltonian vector fields reproduce either the n-th Gel'fand-Dikii hierarchy (for n finite) or Sato's hierarchy (for n = ∞). Within the same framework, it is also possible to recover a class of integro-differential hierarchies involving a finite number of fields, which generalize the Gel'fand-Dikii equations and are equivalent to Sato's hierarchy.