Do non-linear metric theories of gravitation really exist?

Abstract

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.