ar X iv : g r - qc / 9 51 10 27 v 2 9 N ov 1 99 5 Are there metric theories of gravity other than General Relativity ?

Current generalizations of the classical Einstein–Hilbert Lagrangian formulation of General Relativity are reviewed. Some alternative variational principles, based on different choices of the gravitational field variable (metric tensor, affine connection, or both) are known to reproduce – more or less directly – Einstein’s gravitational equations, and should therefore be regarded as equivalent descriptions of the same physical model, while other variational principles (“Scalar–tensor theories” and “Higher–derivative theories”) yield pictures of the gravitational interaction which appear to be, a priori , physically distinct from GR. Such theories, however, are also known to admit a reformulation (in a different set of variables) which is formally identical to General Relativity (with auxiliary fields having nonlinear self–interaction). The physical significance of this change of variables has been questioned by several authors in recent years. Here, we investigate to which extent purely affine, metric–affine, scalar–tensor and purely metric theories can be regarded as physically equivalent to GR. Focusing on the so–called “nonlinear theories of gravity” (NLG theories), which presently enjoy a renewed attention as possible models for inflationary cosmology, we show that if the metric tensor occurring in a nonlinear Lagrangian is identified by assumption with the physical spacetime metric, relevant physical properties (positivity of energy and stability of the vacuum) cannot be assessed. On the other hand, using an alternative set of variables (the “Einstein frame”) one can prove that for a wide class of NLG theories positivity and stability properties do hold. This leads one to regard the rescaled metric (Einstein frame) as the true physical one. As a direct consequence, the physical content of such “alternative” models is reset to coincide with General Relativity, and the “Nonlinear Gravity Theories” become nothing but exotic reformulations of General Relativity in terms of unphysical variables.