Legendre transformation and dynamical structure of higher-derivative gravity

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.